Field Manual - Forgotten Books
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Transcript of Field Manual - Forgotten Books
FIELD —MANUAL
RA I L ROA D EN G IN E ER S ,
J. C . NAGLE,M.A.
,
P rofesso r nf C ivi l Emm ineerina in the Ag ricu l tura land Mecha m ca t Co l lege of Texa s.
SECOND ED ITION,ln
’
E VINED.
SECON D THOUSAN D.
NEW YORK:JOHN WILEY 85 SONS.
LONDON :CHAPMAN HALL , LIMITED .
1 903.
Copy r ight , 1897.
BY
J . C . NAGLE .
ROBERT DRUMMOND ELECT ROTYPER AND PRINT ER, N 1
PREFACE .
EASE of reference and uniform i ty of notat ion are essent ial in a
book that i s to be consul ted in the field. W ith this in m ind an
effort has been m ade in the fol low ing pages to secure a system at icarrangem ent of the subject-m atter and uniformity of term s andno tat ion. Except for a few cases G reek letters have been avo idedand a s ingle letter i s used to designate an angle. In so far as
self-explanatory, so that
the explanat ions necessary in connect ion w ith the problem s havebeen reduced to a m ini m um . Algebraic equat ions stand eac h ina dist inc t l ine, thus rendering them m ore easi ly read.
A know ledge of the elem ents of geom et ry and trigonom etry has
r heen assum ed,and only in the derivation of a few form ulas in
“?connec tion w i th the theo ry of trans i tion-curves w i l l any h igherm athem at ic s be needed. But these form ulas m ay be accepted bythe reader who is unfam i liar w i th the calculus withou t in any
way affec ting h i s abilitv to understand the i r appl ications or tofol low subsequent reasoning.
One can m ost readi ly turn to what he wants in a book after hav6 ing becom e fam i l iar With i ts contents in the c lassroom . K eep ing
ha; this in m ind this book has been Written so that i t m ay be used as
a text as wel l as for reference in the field. Wherever prac t icableKNi
soiut ions to problem s have been g i ven in a rigid, general form ,
S fo llowed by i llustrat ive exam ples,so that the student need not
10 8 8 s ight of the princ iple involved wh i le fo l low ing the solut ionfor a particular case. Wherever approx i m ate solut ions seemedpreferable they have al so been gi ven and thei r l i m i tations pointedFree use has been m ade of the T able of Func t ions of a One
degree Curve,thus reduc ing the labor of field com putations . By
defining the degree of curve w i th reference to short chords foriii
PREFACE ,
sharp curves— and,wi th tab les of Radi i
,Long Chords
,Mid
ordinates, et c based on appropr iate equations— the errors resul ting from assum ing the radius to vary inversely w i th the degreeof curve wi l l generally be found to be qui te sm al l .Chapter I g i ves briefly the general m ethod of m ak ing Re
conno i ssance; Chapter II treat s of P rel im inary Surveys ; whileChapter III relates to Location.
Chapter IV ,on T ransi tion- curves , fol lows the m ethod adopted
by P rofessor Crandal], and enables one to locate the transit ioncurve wi th rigid accuracy where such i s necessary . Approx i m atem ethods are al so given by m eans of wh ich the curve m ay be as
easi ly located as any of the m ore l i m i ted easem ent curves ordinarily m et wi th .
Chapter V ,on Frogs and Swi tches , contains all that i s necessary
for thei r locat ion. The form ulas have been arranged to give thedes i red quant it ies in term s of the frog num ber whenever the re
sult ing equat ions would be eas ier of appl icat ion than the tr igonom etric ones usua l ly gi ven. T he turnout tab les are unusual ly ful land give not only the theoret ical lead but the stub lead as wel l
,
from wh i ch the pract ical lead can be at once found when the
length of sw i tch - rai l i s known.
Chapter V I,on Const ruction,
tel l s how to set slope-stakes,and
gives s i m ple m ethods for com puting areas and volum es ei therdi rectly or by the use of tables . A short table of pri sm o idalcorrect ions is given for end sect ions level , and al so a form ula forthree- level sec tions , by m eans of wh i ch a su i table table m ay be
com puted if des i red.
T he tables at the end of. th is book have been arranged w i th av iew to ease of reference,for
,whatever the character of the text ,
the ch ief value of a field-book m ust depend upon the ease w i thwh ich the tables m ay be consul ted and upon thei r extent and
accuracy . T ab le IX— Func t ions of a One-degree Curve— separates the logar i th m ic func t ions on the one side from the naturalfunct ions on the other and w i l l be of ass i stance in locat ing thesetables . T able XV I— T rans i t ion-curve T able— reading lengthwiseof the page , l ikew i se serves t o separate the trigonom etr ic tablesfrom the m i scel laneous tab les that fo l low .
Som e engineers object to the use of logar ithm ic tables in the
field,but for them the natural func t ions are at hand ; wh ile for
those who prefer logarith m s the fi ve-place tables of logarith m icS ines , cos ines, etc w i l l be found easy to consul t and interpolatebetween.
PREFACE. V
All trigonom etric tables are five-place , and others were carr iedto as m any dec i m a l places as the i r charac ter dem anded.
T ables I , III, IV ,and V have been com puted to agree w i th
the definit ion of the degree of curve requi ring curves sharperthan 7
°
to be run wi th chords less than 100 feet in length ,as
desc ribed in the text . T ables XV I I and XV I I I were al so com
pa ted expressly for th i s book .
T ables V I and XXVII are from elec trotypes fro m t art ’sField Bookfor Civil Engineers and were furni shed by (n ul l St Co .
Electrotypes of Tables II, X ,XII
,X II I
,X IX
,XX
,XXIV
,XXV
,
XXV I, and al so XVI — th i s last being from Crandal l ’s book,
The Transition Curve— were furni shed by John Wiley Sons .Of the o thers
,som e were arranged from standard tables and
o thers adapted in part and extended t o increase thei r usefulness .It w i l l be not iced that vert ical l ines have been om i tted wher
ever pract icable, thus render ing i t easier to refer to the tables .Acknow ledgm ents are due m y assoc iate
,Professor D. W.
$ ‘pence,for aid inm aking the tabular com putat ions and in reading
proof.
J C . NAGLE .
COLLEGE STAT ION , T m s , May , 1897.
PREFACE TO THE SECOND EDIT ION .
IN this edition some of the typographical and other m inor errorsthat appeared in the first edi t ion have been el im inated. T ablesXXV III and XXIX have been added in order to inc rease the usefulness of the book
,and are from elec t rotypes of tables in T i'autwine’s Pocket Book. A suggest ion has been m ade by one who
has had occasion to use the tables qui te freely that T able XIX be
extended so as to gi ve quant i t ies for variat ions of one tenth of a
foot in center heights, but such extension would have inc reasedthe s i ze of the book unduly . When c loser approx i m ations arewanted than are given by T ableXIX the area for the gi ven centerheigh t can be taken from T able XVII and by entering T able XXw i th thi s as argum ent the quanti ty can be at once read off. For
center heights greater than those given in T able XVII we m ay
refer to books devoted exc l usively to earthwork computations .J C . N .
COLLEGE STAT ION , T EXAS , January, 1899.
CONTENTS.
CHAPTER I .
RECONNOISSANCE .
ART ICLE 1 . OBJECT S OF RECONNOISSANCE— HOW MADE .
SECT ION PAGE
1 . Relat ive Im portance of theWork of Reconno issance and Location 1
2 . Ob jec t ofRec onno issance 2
3. The Instrum ents 2
4 . Use of Maps 4
5. Mak ing the Reconnoissance 4
CHAPTER II.
PRELIMINARY SURVEYS.
ARTICLE 2. OBJECT S ; T HE FIELD CORPS ; DUT IES OF T HE CH IEF.
6 Ob jec ts of Prel im inary Surveys7 . T he Exp lora t ion8 . Data Sought in Mak ing Prel i m inary Surveys9 . The F ield Corps .
10. The Ch ief of Party , Dut ies ofART ICLE 3. THE TRANSIT PARTY}
A . DUT IES OF THE MEMBERS .
11 . Co m position of the T ransit Party12. The T ransitm an
13—17 . Other Mem bers of the Party18 . Instrum ents
B . TRANSIT ADJUSTMENT S— THE VERNIER.
19 . K ind of T ransit20 . T o Adjust the Plate Levels21 . Paral lax . .
22. T o Adjust the L ine of Co l l im ation.
23. T o Adjust the Standards .
CONTENTS.
SECT ION
24 . T o Adjust the Level on Telescope25 D irec t and Retrograde Verniers26. T he Least Count of a Vernier
T o .Read a Vernierc . ACCESSORIES .
The Gradienter.
28 . Descr ipt ion and Method ofUsing Gradienter .The S tadia , or Telem ete‘r.
29 . Princ iple of the Stadia30 . Form ula for L ine of Sight Horizontal31 . Form ulas for L ine of S ight Inc l ined
The Instrum enta l Constant , T o Find33. Reducing the Notes
D. FIELD-WORK .
34 .
35. HUb S OP PlugS o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o
36. Reference-po ints37. A l ignm ent38 . Form ofT ransi t Notes39. Stadia Methods for Prel im inary
E . OBST ACLES ix TANGENT .
41. To Pass an Obstacle by Means of Para l lel42. T o Pass an Obstac le by Angular Deflec t ions43. T o Measure across a River
ART ICLE 4 . T HE LEVEL PARTY.
44. Make-up and Instrum ents45. Work of the Leveler46. Work of the Rodm an
ADJUSTMENTS OF THE LEVEL.
47. To Adjust the L ine of Com inat ion
48 . T o Adjust the Level-bubb le49 T o Adjus t theWyes .
B. THEORY OF LEVELING .
CONTENTS.
SECT ION PAGE
56. T he Level Notes 28
57 . Prec autions when Using Level 29
58 . 29
ART ICLE 5. TurnTOPOGRAPHIC PARTY.
59 . Instrum ents Used; Area to be Mapped60 . Methods ofRecording Data61 . T opographers‘ F ield-sheets62. Use of the Slope-level .63 . C ross-sec tion Rods64 . The T ransit and Stadia in Topographica l Survey ing .
ART ICLE 6. PRELIMINARY EST IMAT ES.
66. Map of Prel im inary L ines . 32
67 . T he Profi le . 33
68 . Prel im inary Est im ates ofQuant it ies 33
69 . Report of the Locating Engineer 34
CHAPTER III.
LOCAT ION .
ART ICLE 7. Paom crm o LOCATION.
70. Problem s Involved in the Paper Locat ion7 1 . H ints Regarding Methods of Projec ting the L ine72. T he Curve-protrac tor73. Work in the Field.
ART ICLE 8 . SIMPLE CURVES .
A. DEFINIT IONS AND FORMULAS .
74 . Definitions
75. T o Find the Radius R, the Degree ofCurve Being K nown76. T o F ind the Length of Curve77. The Func t ions ofa One-degree Curve .
79. To F ind D, R and 0 Being K nown80. To Find the Tangent Distance
_T , I andR Being K nown
8 1. T o F ind R, G iven 1 and T
82. G iven l and D, to Find the Long Chord83 . Ordinates from Chord84—86. T o F ind the Ex terna l E87 . T o F ind R, E and I G iven88 . T o F ind T , E and 1 G iven89 . To F ind the Deflec t ion Offset from Chord Produced90. T o Find the T angent Deflec t ion Offset91 . The Sub -tangent ia l Deflec t ion Ofi'
set . .
92. T o F ind the Tangent Offset z93. D ifference in Length of A re and Long Chord
CONTENTS.
B . LOCAT ING SIMPLE CURVES.
SECT ION PAGE
94 . T o Locate a Curve with the Cha in by Ofi sets from Chords Produced 55
95. T o Locate a Curve by Ofifsets from T angent 57
96. T o Locate a Curve by Offsets from a Long Chord 58
97 . T o Locate a Curve w ith T ransit and Cha in . 59
98 . T he Index -angle 60
99 . Subdeflec t ion-ang les 60
100 - 101 . T ransit Notes 61
C . OBSTACLES.
102 . T o Pass an Obstac le on a Curve103. T o Locate a Curve when the R C . is Inaccessib le104 . T o Pass to Tangent when the R T . is Inaccessib le105—107 . T o Pass a Curve through a G iven Po int .
108 . T o Locate a Tangent to a Curve fro m an Outside Po int109 . T o Run a Tangent to Two Curves of Contrary F lexure
D . CHANGE OF LOCAT ION .
110 . T o Locate a Curve Para l lel to a G iven Curve 73
111 . T o Change B C . in Order'
to Make R T . Fa l l in a Paral lel T angent . . 74
T o Change R and R C . t o m ake P . T . Fal l in Para l lel Tangent. onSa m e Radial L ine
113 . To F ind Change in R C . or R for a G iven Change in I .
114 . Required t he Change in R C . and R for a G iven Change in I , the1 15. T o Find New Radius for a Given Change in T116. T o Find New R to Connect P C . with a Para l lel T angent
ART ICLE 9. COMPOUND CURVES .
A . LOCAT ION PROBLEMS.
117. G iven Both Tangents and One Radius , t o Find the Other Radius 80
1 18 . G iven One Radius. the Long Chord and the Ang les i t Makes withT angents , t o Find the Other Radius and Centra l Angles
119. G iven the Rad i i and Centra l Ang les . t o F ind the T angents , the LongChord. and the Angles i t Makes w ith T angents
120. Given the Long Chord and Ang les Made w ith T angents , t o F indBoth Radi i when Com m on Tangent is Paral lel t o Long Chord 83
B. OBSTACLES.
121. To Loc ate Second Branch whenR C. is Inaccessib leC . CHANGE OF LOCAT ION .
122. TO Com pound a Sim p le Curve so P . T . shal l Fa l l in a Paral lel Tan123 . To Find Change in Necessary t o Make P . T . Fa l l in a Par
a l le l T angent .
124 . T o Change P . C .C . and Sec ond Radius so P . T . sha l l Fa l l in a Par
a l lel Tangent , on Sam e Radia l L ine
CON T ENTS.
sm c'ro m en;
25 To Change F .C .C . and Second Radius to Cause P . T . t o Fa l l at a
New Po int in Sam e Tangent 9 1
126. To Subst itute a Three-c entered Com pound Curve for a S im ple One. 94
127. TO Substitute a Curve for a Tangent Uniting Two Curves 95
ART ICLE 10. TRACK PROBLEMS .
28 Reversed Curves, Where to Use129 . T o Connec t a Located Curvewi th an Intersec ting Tangent .130 . T o Locate a Y131. A Reversed Curve between Para l lel Tangents13° A Crossover b etween Paral lel T rac ks when a F ixed Length ofTan
gent is Inserted .
133. A Reversed Curve with Unequa l Ang les134. A Reversed Curve between Fixed Po ints135. T o Connec t Two D ivergent T angents by a Reversed Curve136. T o Change P .R.C . so R T . shal l Fa l l in a Para l lel T angent137. To Find theRadius of a Curved T rack .
CHAPTER IV .
TRANSIT ION -CURVES.
ARTICLE 11. THEORY OF THE TRANSIT ION-CURVE .
138 . Eleyat ion ofOuter Rai l on Curves .
139 . Requirem ents of the T rue T rans ition-curve140 . Notat ion Em ployed .
141 . Equation of T ransit ion-curve142 . T ransit ion-curve Angle, I143. Coo
’ rdinates of Po ints144 . Deflec t ion-ang les145. Exp lanation of T ransit ion-curve Tab les .
146. T o Unite the Branches of a Com pound Curve by a T rans ition147. Length of T ransit ion-curve to be Taken
ART ICLE 12. FIELD -WORK .
A . FIELD FORMULAS.
148 . When to Use the Sim p lified Form ulas149 . Sim p lified Form ulas for T ransition- curves150 .
151 . Com pound CurvesB. SET T ING OUT T RANSIT ION ° CURVES .
153 . Locat ion by Offsets .
154 . Location by Deflec t ion-angles155. Form of T Iansit Notes for T rans i tion-curves. .
CON TENTS.
ART ICLE 13 . T RANSIT ION CURVE PROBLEMS .
SECT ION
156. T angent D istances and Externa l for Equa l Offsets .
157 . T angent D istances , Offsets Unequa l .158 . T rans it ion-curves Inserted without Changing the Vertex of Cir
cula t Curve159. T ransition-curves Insertedwith Least Deviation f1 om Old T rack" . 133160 . T r -ansit ion curves Inserted at Ends of Long C ircular Curve, Centra l Port ion Undisturbed161 . T ransition-curve Inserted at P . by Changing Radius of Sec ond
PAGE
162. T o Insert Transit ion-curves a t the Ends of Two C ircular CurvesUnited by a. Com m on T angent
163 . T o Unite a T angent and C ircular Curve when the Ofi set Cannot beD irec t ly Measured
164. Inserting T ransit ion-curves in Old T rack165. Rem arks on T abular Interpo lations
CHAPTER V .
FROGS AND SWIT CHE S.
ART ICLE 14 . T URNOUT S .
A. TURNOUT S FROM STRAIGHT LINES
167 . T o Find the Lead, l , and Radius, R, in Term s of the Frog Num ber.
168 . G iven R and g , to F ind N , l , and Frog-ang le, F
169 . To Find Theoret ic Length of Switch-ra i l170. To F ind Lead and Num ber of Crotch - frog for a Doub le Turnout t o
Oppo si te S ides ofMa in T rack171. T o F ind Turnout Radius and Lead of Crotch-frog in Term s of
O O O O O O O O O O O O O O O O O O O O
172. T o F ind Radius of Curve from Po int of M iddle Frog to Po int ofMa in Frog, G iven N , , N . and N
173. Doub le Turnout t o Sam e S ide ofMain T rac k174 . T o Find Radius of Curve b etween Frog-po ints for a Doub le Turn
out t o Sam e Side of Ma in T rac k175. T o Unite Ma in T rac k with Siding . Reversing Po int Opposite Frog 152
176. T o Lay Out a Ladder-trackB . TURNOUT S FROM CURVES .
177 . To F ind Lead and Radius for Turnout t o Concave Side of MainL ine
178 . T o F ind Lead and Radius. T urnout to Convex S ide179 . T o F ind Theoretic Length of Switch-ra i l .180 . T o Unite Ma in T rack W i th a Concentr ic Siding
CONTEN TS.
0 . THE STUB LEAD.
SECT ION
18 1 . Definit ions
182. G ivenN , t , and g, to Find the Stub Lead183 . T urnout T ab le and Explanation184. T o Stake Out a Turnout185. Carving Rails .
ART ICLE 15. Caossovm as.
186. Crossover between Para l lel Stra ight T racks. 8. Tangent betweenFrog-po ints
187 . A Crossover in t he Form of a Reversed Curve188 . A Crossover with F ixed Length of Interm ediate Tangent189. A Crossover between Curved Ma in T racks
ART ICLE 16. CROSSING-FROGS AND CROSSING-SLIPS.
A . CROSSING -FROGS.
191. Length of Rai l Intercepted between Two Intersec ting Straight192. Ang les ofa Set ofCrossing frogs , One T rack Curved193. Ang les ofa Set ofCrossing -frogs. Both T racks Curved.
B . caose e -sm p s .
195. Length and Radn of Sl ip -ra i ls, Both T racks Stra ight196. Length and Radi i of Sl ip-ra i ls, One T rack Curved197. Length and Radi i of Sl ip -ra i ls , Both T racks Curved]
CHAPTER VI.
CONSTRUCT ION .
ARTICLE 17. DEFINIT IONS ; GENERAL CONSIDERAT IONS VERTICAL
CURVES ; ELEVAT ION OF OUTERRAIL .
199. The D ivisionEng ineer200 . The Res ident Eng ineer201 -204 . Definit ions
205. To a“ind the Grade-point , Long itudina l Slope Uniform "
206. V ertica l Curves207 . E levat ion ofOuter Ra i l on Curves .
208 . Easing Grade on Curves .
ART ICLE 18 . EARTHWORK .
A . SETT ING SLOPE-STARES.
209 . The D istance Out for Level Sec t ions210. T o F ind Posit ion ofSlope-stakes for Surface Inc l ined211 . Cross ~ sec tion Notes212. Irregular Sec tions213. Stak ing Out Openings
CONTEN Ts
SECT ION
214 . Manner ofMark ing Stakes .
215. Shrinkage— Growth216. Borrow-pi ts, Dra inage of,
B. AREAS OF SECT IONS.
218 . Areaof~
T hree- level Sec t ion .
219. Area ofFive- level Sec t ion .
220. Genera l Form ula for Areas221. Exp lanation of T ab le of Areas of Leve l Sec tions and the Three
C . VOLUME OF EART HWORK .
222. Where Cross-sec tions should be Taken223. Vo lum e by Averag ing End Areas .
224 . T he Prism o ida l Form ula226. The Prism o ida l Correc t ion227 . Com putation of Vo lum es when Passing from Cut t o F i l l . .
228 . Use of Tab les ofV o lum es in Mak ing Prel i m inary Est im ates229 . Side D itches . .
230. Earthwork on Curves231 . Overhaul
ART ICLE 19. GRADE AND BALLAST STAK ES , CULVERTS , BRIDGES,AND T UNNELS .
232 Grade and Center Stakes233. Ba l last-s t-akes .
235. Openings of, for Culverts. T restles, etc236. Bridge P iers and Abutm ents
ART ICLE 20. MONTHLY AND FINAL EST IMATES.
238 . Monthly Esti m ates239. Measurem ents for Earthwork240 . 0 1assificat ion of Earthwork2 11. The Progress 207
242. Masonry Esti m ates .
243 . Bridge Est im ates .
244. T rac k Materia l245. B lank Esti m ate Sheets
TABLES.
Tab le Showing Length ofT ransition-curve to beT akenTab le ofV a lues of g Vgt for Stub LeadT urnout Tab leTab le of Co rrec tions for Vert ica l Curves
CONT ENTS.
T ab ie of E levation ofOuter Ra i l on CurvesT ab le of Prism oi’da l Correc t ions for Leve l Sec t ions
I . Radi i o f CurvesII. M inutes in Dec i m a ls of a DegreeIII . T angentia l OffsetsIV .
V .
Long Chords and Ac tua lMid-ordina tes to Long Chords
VI. Logarith m s of Num bers
VII . Logarithm ic S ines and CosinesVIII .
IX .
X .
XI .
XII .
XIII .
XIV .
XV
XVI .
XVII .
XV III .
XIX .
xx .
XXI .
XXII .
XXIII.
XXIV .
Logar i thm ic T angents and CotangentsFunc tions of a One-degree CurveNatura l Sines and CosinesNatura l Secants and CosecantsNatura l Tangents and Cotangents '
.
Natura l Versines and ExsecantsCo6rdinates for T ransi tion-curvesDeflec t ion-angles for T ransition-curves. .
T ransition-eurve Tab leA reas of Level Sec tionsCorrec t ions for T hree-Ievel GroundCub ic Yards per 100 ft . in T erm s of Center HeightCub ic Yards per 100 ft . in T erm s Sec tiona l AreaRise per M i le of V ar ious GradesS lopes for T opog 1 ap hyMa te1 ia1 Requi1 ed for One M i le of T 1a . .c kMutua l Conversion ofFeet and Inches into Meters and Cent i »m eters ” 388
XXV . Mutua l Conversion ofMi les and K i lom eters 389
XXV I . Length of 1’ A 1 0 ofLat itude and Longitude 389
XXVII. T rigonom etric and M iscel laneous Form ulas 390
XXV III . Square Roots and Cube Roots of Num b ers from . 1 t o 28 395
XXIX . Sq11ares. Cubes , Square Roots , and Cub e Roots , o f Num bersfrom l to 1000
2 A F I ELD -MANUAL FOR RAILROAD EN GIN EERS.
grasp and weigh all the com plex features of the question. A
passing reference only can be made to i t in this l i ttle vo lu m e,
which is intended to furnish hints and aids to the better executionof the second part. For the benefi t of the beginner who has todo w i th the location and constru c t ion a few definit ions and hintsrelating to reconnoissance wi ll b e gi ven before go ing on to the
special problems arising in the wo rk of the rai l road engineer.
2 . T he Reconno i ssanc e i s a rapid, general survey of the area
through which the proposed rail road m ust pass,m ade only with
such instrum ents as can be easi ly carried, and—which should ena
ble the engineer to rest ric t the m ore accurate instru m ental workthat fo llows to one or two general l ines . T he ti m e required forthis part of the work wil l in general be only a smal l fraction of
the ti m e consum ed in location, involving the service of very fewm en:yet there i s no par t of the work m ore rapidly and im
p roperly ( lone— no t always because the engineer in‘
charge underest im ates i ts i m portance. but because he i s not usual ly al lowedsufficient ti m e in which to study thoroughly the area under consideration.
Properly the reconno i ssance inc ludes the determ ination of the
term inal po ints of the road,but the locat ing engineer is usually
rel ieved from the necessity of selec ting these points , and thequestion reduces t o that of finding the best available line which
adm its of being built, m aintained,and op eratedfat the least cost
between two given points.
T he reconno issance m ust b e m ade over an area— not a l ine or
l ines . Even what seem s the m ost unprom i s ing port ion shouldbe careful ly studied
,for the engineer can never be sat isfied he
has selec ted the best route unt i l he has conv inced h i m self by careful study that a ll others are inferio r . T oo m uch haste on reconnoissance m eans e i ther a poor l ine or a m uch greater expenditureo f t i m e and m oney on the prel i m inary . N o am ount of notes ortopography can take the place of an int i m ate personal knowledgeof the problem s to b e encountered,
and hence the reconno i ssanceand prelim inary survey should be m ade by the engineer who i st o locate the road.
3 . T he Inst rum ents needed wi l l rarely be m ore than a pocketcom pass , hand- level
,anero id barom eter
,field-
glasses , and som e
ti m es a pedom eter or an odom eter.(ft ) T he P o ck et - c o m pas s i s used to obtain the m agnet ic bearings of l ines and the angles they m ake w i th each other .
RECONN OISSANCE .
(b) T he H and-lev el enables one to obtain differences of ele
vation between po ints not far apart .(0) T he Aneroid Barometer gives approxim ate heights of themercury colum n,
and serves to rough ly determ ine the difierence
of elevat ion of g iven po ints . In addi tion to the scale givingreadings in inches , it should have al so a scale graduated to g i vereadings in feet . If two anero ids
,wh ich have been previously
com pared,are read sim ul taneously
,one at each of the po ints
whose difference of elevat ion i s desi red,or if the same anero id i s
read at each success ively at a short interval of t im e,during wh ich
the atm ospher ic pressure has not sensibly a l tered, we m ay find
the difference of elevat ion by the form ula !d 60000 (log H log 71) (1 M ),900
in Which d i s the difference of al t itude in feet , H and h the
barom etric readings in inches— the logarithms being of the com
m on or Briggs k ind, T and t the tem peratures of the two stat ionsin Fahrenhei t degrees .If the sum of the tem peratures, T t, i s taken as formula
(1 ) reduces tod 63000 ( log H log h).
EXAMPLE .—The reading of the barom eter at the foot of a
mountain i s inches , and at the top inches . Requiredthe height of the m ountain.
By d 63000 ( log log 2071 feet .T he effec t of tem perature on the m etal of the instrum ent
should be considered in the barom etric form ula when very pre~
c ise work is to be done but th i s correction, being sm al l , m ay be
neglec ted in the rough work of reconno i ssance, particularly sincethe m akers of the instrum ent construct i t in such a way as to
com pensate,as c losely as possible,
for such changes of tem
perature.
(d) T he P edomet er i s an inst rum ent which au tom at icallycounts the num ber of steps m ade by a person when the instru
m ent i s attached to h i s bel t ; then, know ing the average lengthof step
,the distance passed over can be readi ly com puted.
T he Odom eter registers the nu~
nber of revolutions of a wheelt o which it i s attached,
and the num ber of revolut ions m ultipl iedby the c i rcum ference of the wheel g i ves the space passed over .
See P ly m pton’
s Anero id Barom eter, {1 38. for form ula
4 A FIELD-MANUAL FOR RA ILROAD EN GIN EERS.
4 . T he M ap .— Before beginning the reconnoi ssance the engi
neer should prov ide him self w i th the best ava i lable m ap of the
region to be t raversed ; if th is i s a topographic one,he can at
once determ ine from it the l ines that are l ikely to justify an
exam ination '
and even if i t is only a sketch -m ap ,he can get
material assistance by observ ing the courses of the stream s and
rem em bering that thei r posi t ions indicate the relat ive elevat ionsof the port ion of the region through which they flow . T hus thelarge stream s fo l low the l ines of least elevat ion, and the m annerin wh ich the lateral stream s uni te wi th the princ ipal one indicates the general t rend of the terrain. T wo st ream s flowingnearly para l lel approach or recede from each other according as
the intervening land di m inishes or increases in al t i tude. Two
stream s flowing away from each other on opposite s ides of a
div ide,and hav ing thei r source therein,
approach each o therc losest at the po int of least elevat ion,
and indicate the posit ion of
a pass or the lowest point of the di v iding ridge. T he study of
any good contour m ap covering sufficient area will i l lustrate thelaws governing the courses fo l lowed by stream s .The elevat ions of a few correctly m apped po ints , when obta in
able,from the m ap or otherw i se
,serve as a guide in tentat ively
fixing on the m ax i m um gradient to be em ployed and the am ountof developm ent needed.
A sk i l lful eng ineer w i l l thus be enabled to
.
project h is l ineswi th suffi c ient accuracy to enable him to select on the ground them ost feas ib le route or routes for h is prel im inaries in the leastpossible t i m e. He should guard aga inst the conv ict ion
,however
,
that i t i s unnecessary for him t o look el sewhere than a long the
projec ted routes for the inacc uracies of the m ap ,local pecul iar i
t ies , the nature of the excavation and em bank m ent , the num berand cost of bridges and o ther m echanica l structures ,— all thesem ay consp i re t o m ake the m ost prom is ing m ap
- l ine infer ior tosom e other whose advantages have to be sought for on the
ground.
5. Hav ing tentat ively dec ided on the l im it ing grades and cut
vature t o be em ployed,the engineer goes careful ly over the
ground , exam ining the enti re area that seem s l ikely t o afiord
passage,in order to determ ine whether a sui table l ine m ay be
secured for the grades and curves previously assum ed. W i th hispocket-com pass he takes the bear ings of l ines , and by m eans of
the hand-level and anero id determ ines differences of elevat ion.
RECONNOISSAN CE.
Di stances are esti m ated by the eye,paced,
and the count takenfrom the pedom eter
,or
,if the country adm its of the use of avehic le
,taken from the odom eter readings . If a wel l -ga i ted
saddle-horse i s used, very good resul ts m ay be gotten by t iminghim ,
or by the use of the pedo m eter if h is str ide i s uniform .
But in all cases m uch dependence must be placed on the ab i l i tyto est im ate w i th the eye difi erences of elevat ion and distances .T he abi l i ty to do th is wi th even reasonable accuracy com es onlyfrom long pract ice and careful observat ion, even to the mostgifted in th is respect . New and unexpected condi t ions som et im es deceive even the m ost pract iced eye,
but under ordinarycondit ions alm ost any one can tra in hi s eye to estim ate hor izonta ldistances fai rly wel l . V ertical heights are m ore deceptive , pos
sib ly because we have less pract ice in th i s l ine,and the m ind
seem s natnrally to exaggerate the vert ical as com pared wi th thehorizontal ; prac t ice, however, wi l l enable us to m ake al lowancefor the natura l tendency to overest imate heights and slopes .The ground should be gone over in both di rec tions , for the ap
pearance m ay be qui te different when approached from differentquarters . Rul ing points , such as a pass in the m ountains , thecrossing of a large stream ,
or a town or c i ty through wh ich theroad m ust be bui lt , serve to reduce the problem to a number ofspec ial ones
,each hav ing i ts own sol ut ion.
In a m ounta inous region offering a l im ited number of possibleroutes
,but heavy construction work , i t m ay often happen that
the locat ion of a l ine i s a much less difficult operation than in an
open,rol l ing country offering a score of possible l ines
,between
which the engineer making the reconno i ssance must dec ide;selecting only those that in his j udgment seem to just ify an
accurate instrum ental survey.
T he engineer m ust keep constantly in mind all the factors ofthe general problem of econom ic locationand m aintenance,
and
successful operat ion of trains . One l ine m ay cost m ore for construc t ion and m aintenance than another
,but less for operation ,
or m ay inv ite less t raffi c . In all cases , however , the quest ionof grades , curvature,
length of l ine, earthwork , and m echanicals truc tures are the contro l l ing elem ents to be cons idered.
Hav ing dec ided upon the route or routes over which to run
prel i m inaries , these are m arked on the m ap ,and the engineer ing
party organi zed and put in the field,with all the necessary
instrum ents.
CHAPTER II.
PRELIMINARY SURVEYS .
ART ICLE 2. OBJECT S; T HE FIELD CORPS ;DUT IES OF THE CH IEF .
6. T he Objec ts of the prel im inary surveys are to secure all the
data necessary to determ ine which one of the routes selected on
reconno i ssance i s the m ost feasible,all things considered,
and the
approxi m ate cost of construct ion. In rough country it wil l beeconom ical to m ake two
,or even th ree
,surveys over the route se
lected for location before beg inning t o place the l ine in the pos i t ioni t i s finally to occupy. T he first of these is often omitted
,and is
cal led an explorat ion- l ine i t wi l l frequently save the m ak ingof the m ore expensive “ prel iminary over one or m ore of the
routes .
’7. T he E x plora t ion-line m ay be made w i th ei ther transi t or
com pass , and consists of a rap idly run l ine,m ade for the purpose
of determ ining the m axi m um curvature and gradients with whichto project the prel i m inary . It w i l l not be necessary t o m ake a
detai led study of the region at th is t i m e , the di stances and elevat ions
,wi th such sketch topography as m ay be eas i ly taken
,being
all that i s needed. T he m agnet i c bearing of l ines i s taken bythe com passm an
,and the chainm en al ign each other with the flag
set by the fiagm an. As the progress of the level party w i l l beslower than that of the com pass party
,i t w i l l be econom ical t o add
an extra_rodm an
,and som et im es a recorder . T he com passm an
m ay sketch in the features adjacent to the l ine wh i le wa i t ing forh i s chainm en
,who m ay be ei ther in front of or beh ind the com
pass .T he stadia m ethod of surveying— to be spoken of later— wouldseem to offer except ional advantages for th i s work— only three or
four m en being needed in addi t ion to the ch ief. W i th it , by sett ing the transi t over al ternate stat ions , very rapid progress m ay be
m ade, and obstac les avo ided w i th as m uch or greater ease thanwith the com pass .T he exploration- l ine wi l l m ore than pay for i tself in showing
PRELIM INARY SURVEYS.
what routes i t wi l l be unnecessary to m ake prel im inaries over,
and in indicating the m ost feas ible one. It should be run over allthe routes selec ted on reconno issance.
8 . T he P rel im inary Surv ey fol lows the exploration,or
,when
this i s om i tted, com es next after the reconno issance . It m ay ,w i th
advantage,be m ade in two parts— first and second prel i m inary.
It i s m ade wi th such instrum ental accuracy as the nature of the
case m ay dem and,suffi cient data being obta ined to determ ine the
best l ine on wh i ch to locate and the approx im ate cost of construet ion. T he rap idi ty wi th wh ich th is work can be done wi l l dependon the care wi th wh ich the reconno i ssance was m ade. T he pre
l im inary l ine should approxim ate,as c losely as the eye can deter
m ine,to the posi tion the located l ine should occupy
,and form s the
base on wh ich the topographic work rests . In reasonably easycountry , where explorat ion- l ines have been run
,one prel im inary
should suffice for each route,but indifficul t regions i t w i l l be best
t o run a second prel i m inary . If portions of the route are easy , followed by difficult parts , i t w i l l often be suffi cient to back upand re-run the difficult portion unt i l a reasonably sat i sfactory l inehas been obta ined.
9 . T he F ield C orp s consists of a chief of party,t ransitm an
,
leveler , rodm an ,two cha inm en
,rear rodm an or
“ back -flag,
”
stakem an,and two or m ore axem en. If a topographic party is
added,as i t should be in any but the easiest country, there wi l l be
al so a topographer wi th two or m ore ass i stants . A cook and
team ster wi l l be needed w i th the cam p outfit .
T he corps i s usual ly di vided into the fo l lowing part ies(a) T 11E T RAN SIT PARTY.
(D) T HE LEVEL PARTY.
(0 ) T HE TOPOGRAPH IC PARTY.
1 0 . T he C hief of P ar ty receives h i s orders from the chief engineer, or such other officer as m ay be in charge, di rects the m o
t ions of the surveying corps , and i s responsible for thei r conduc tand progress . He prov ides accom m odat ions and suppl ies
,pays all
expenses , taking receipts or vouchers for all outlays - in dupl icate when requi red. In the less th ickly populated sec tions hem ust prov ide tents , wagons , “
cook , and all necessary cam ping outfit
and suppl ies . He must di rec t the field operations in person,keep
ing in advance of the transi t , establish turning-po ints or hubs,
and direct the t rans it m an in the proper course. He should keep
8 A F I ELD -MANUAL FOR RAILROAD EN GINEERS.
a record— or di rect the trans i tm an and topographer to do so— of
the character of earthwork likely t o be encountered,the placeswhere dra ins
,culvert s , br idges , cat t le-
guards , etc . ,are needed;
the nature of m aterial for em bankm ent,p i l ing,
etc . ,adjacent t o
the line the probable am ount of c learing and grubb ing,and all
other features l ikely to affect the cost of construct ion. He shouldsee that the nam es of propert y owners and res idents along thel ine and the posi t ions and bearings of property lines
,
' whenposs ible
,are noted.
He should have authority t o dis charge assi stants— except transi tm an leveler , and topographer— whose serv ices are unsat i sfactory ,
and in m any cases it wi l l be best for him t o have ent ire contro l,
engaging or discharging any m em ber of the corps as c i rcumstancesm ay require.
ART ICLE 3. THE TRAN SIT PARTY.~
A. Duties of theMem bers.
1 1 . The T rans i t P arty should consi st of a transitman, headchainm an , rear chainm an , rear fiagm an
,stak em an
, and as m anyaxem en as m ay be requ i red— rarely less than two even for opencountry .
12 . T he T rans i tman cares for h i s inst rument , keep ing i t in ad
just m ent ; di rec ts the cha inm en into l ine; notes the angle betweensuccessive tangents as read on p lates ; notes also the bear ings oftangents , of highways , stream s , and property l ines (on locat ion) ,w i th the plus at which the l ine c rosses them . If there i s notopograph ic party he m ust m ake sketches , on the right -hand pageof note book , of the surface features adjacent to the l ine; thered l ine down the m iddle of page represents the trans i t l ine,whether st raight , broken,
or curved,to wh i ch the sketches are
adjusted. He must see that the axem en keep in line, in orderthat no unnecessary chopp ing m ay be done. Large trees needrarely be fel led on prel im inary, even when a given general coursehas t o be fol lowed
, for sm al l angles m ay be turned t o avo id them ,
the deflect ions t o r ight being m ade to approx im ately balance thoseto left .XVhen the chief of part y is absent the trans itm an i s ranking
m an, and wi l l take tem porary charge.
1 3 . T he H ead C hainm an carries a range-pole or“ flag ,
and
drags the chain, which he m ust see i s straight and hor i zontal
1 0 A FI ELD -MANUAL FOR RAI LROA D EN GIN EERS.
scope,a vertical c i rc le, a level on telescope, stadia w ires ,
and a
gradienter ; the so lar attach m ent w i ll rarely be needed.
2 0 . T o A djus t the P la te L ev els .
—T he ax i s of the instrum enti s set at right angles t o the plates by the m anufac turer
,so thatwhen the ax i s i s m ade vert ical the plates w il l be ho rizontal .
In m aking adj ustm ents rem em ber that a com p lete Teversal
always doubles any existing error .
P lace the bubble- tube paral lel to a diagonal pai r of level ingscrews , and bring the bubb le to the centreb fsits run. Revo lvethe instrum ent 180°
on the vert i cal ax i s,and the level - tube wi l l
be paral lel to the sam e pa i r of level ing-screws as before,but
reversed. If the bubble has m oved from i ts central pos i t ionbr ing i t haZf-way back by m eans of the capstan-headed screws at
the ends of the tube. Relevel and repeat unt i l the bubble rem a insat the centre after reversal . Do the sam e for the . o ther bub b le.
Both bubbles should rem ain at the cent res of thei r tubes during a
com plete reversal .2 1 . P arallax i s an apparent m ovem ent of the cross -w i res withrespec t t o the object sighted when the eye i s m oved from s ide t os ide of
o
the eyep iece,and shows that the i m age does not fal l in the
plane of the c ross-w i res . In prec i se m easurem ents i t should berem oved before m aking an observat ion wi th the telescope. T o dothi s , first br ing the c ross-wires clearly into v iew . when the objectglass i s turned towards the sky, then,
when s ight ing an object ,note if there i s any relative m ovem ent of cross -w i res and i m agewhen the eye i s m oved from side to s ide at the eyep iece if therei s , refocus the object-glass unt i l th is m ovem ent disappears .
2 2 . T o A djus t the L ine of C ol l i m at ion i s to m ake the l inejo ining the intersect ion of cross-w i res and opt i cal center of objec
t ive describe a plane perpendicular to the horizontal axis of instrum ent .FIRST METHOD .
— Level the instrum ent and c lam p the m ovem ents on vert ical axis . Sight som e wel l -defined object di stantabout the length of an average sight , and in the sam e hor i zontalplane as telescope; Reverse the telescope on i ts horizontal ax i s
,
and fix a po int about as far from instrum ent as first po int,and in
the sam e hor i zontal plane. Revolve the instrum ent on i ts vert i calax is and s ight the first po int ; then reverse the telescope and noteif l ine of s ight cuts t he second po int . If not
,loosen the capstan
headed screws holding c ross -w i re r ing and m ove the vertical wi re
PRELIM INARY SURVEYS.
over onefourth the apparent error— s ince there were two reversal s— rem em bering that the im age of the c ross -wires is inverted,
wh i lethat of the objec t appears in i ts true posi tion. T est by repet it ion.
SECOND MET HOD .
— If the l i m b graduat ions can be rel ied on
they m ay be used in adjust ing the vert ical w i re. Wi th the instrum ent level sigh t a wel l-defined po int , then revolve 180° by vern ierplate, reading both verniers ; reverse telescope,
and note if l ine of
sight cuts the po int . Ifnot , correc t one ha éf the apparent error bymov ing diaphragm ; then test by repet i t ion.
T he m anufacturers adj 11st the object -glass sl ide so that the ob
ject ive t ravel s in the telescope ax is,and this adjustm ent i s not
l iable to ser ious derangem ent . It is wel l,however
,to som eti m estest by adjust ing the l ine of co l l i m at ion for both near and di stant
objects . Ifno t correc t for both , m ove the r ing wh i ch gu ides therear end of object-glass sl ide unt i l the adjustm ent is correct forboth pos i tions .Next m ake the vert ical w i re vert ical by not ing if i t co inc idesth roughout i ts length with a plum b- l ine
,or by observ ing if it de
viates from a po int, onWh ich the intersect ion has been fixed,when
the telescope i s elevated or depressed. Any erro r i s corrected by
turning the r ing after sl ightly loosening the screws holding’
it .
T he horizontal w i re should also be adjus ted so that the intersect ion of the c ross -w i res w i l l be in the ax i s of the telescope if
thetrans i t is t o be used as a level ing instrum ent th is adjustm entis essential .Drive a stake c lose to the instrum ent , and w i th the telescope
c lam ped as nearly hor i zontal as can be conveniently done read a
rod held on top of the stake about 300 feet distant,and in l ine
with first stake and instrum ent,drive a second stake and read the
rod on it . Revo lve 180°
on vert ica l ax i s,reverse the telescope and
br ing the horizontal w i re to the form er reading when the rod is
held on first stake if the reading on the second stake i s not thesam e as before
,correc t one ha lf the apparent error by m ov ing the
c ross-w i re r ing. Repeat as a test . The vert ical w i re should aga inbe tested lest the m ovem ent of the ring m ay have caused i t tochange.
2 3 . T o Adjus t the Standards i s t o m ake the plane desc ribedby the l ine of col l i m at ion vert ical . Set up the trans i t about as farin front of som e high bui lding ,
or o ther tal l objec t , as the highestpo int that can be s ighted i s above the base. Level the inst rum entand fix the intersec tion of the c ross -wi res on the h ighest po int that
12 A FIELD -MANUAL FOR RAILROAD ENGIN EERS.
can be easi ly sigh ted. Depress the telescope and fix a point nearthe base of the bui lding at about the height of the telescope. Um
c lam p and revo lve on the vert ical ax i s unt i l the telescope reversedcuts the lower po int . C lam p the plates and rai se the telescopeunti l the cross-w i res are at the height of the upper po int . If theycut i t the standards are in adjustm ent . If they do not
,bring
them halfway back by m eans of the adjustable screws at the topof one of the standards . Repeat as a test .2 4 . T o Adjust the L ev el on T elesc ope i s to make the bubblestand at the center of i ts run when the l ine of sigh t i s horizontal .Bring the telescope as nearly hor i zontal as m ay be convenient , andt ake readings on the tops of two pegs in the sam e vertical planewi th
,and equidi stant from
,the instrum ent— say 300 feet . T he
difference of readings w il l equal the difference of elevation of the
pegs ; th is difference m ay be obtained w i th the wye-level if preferred.
Move the instrum ent t o a point beyond one of the pegs and inl ine with both . Set up as c lose to nearer peg as convenient , butnot so c lose that the rod cannot be easi ly read. Bring the telescope as nearly horizontal as possible, and read on both pegs . If
the difference of readings equal s thei r difference of elevat ion thel ine of s ight i s hor i zontal , and the bubble m ay b e brought to thecenter by m eans of the adjustable screws attaching the leveI- tubet o the telescope. If th i s i s not the case
,we m ust set the telescope
so the reading on second peg equal s the reading on first peg plusthe difference of elevat ion ; then read again on first peg and pro
c eed as before unt i l the condi tion i s sat isfied. Or we m ay proceedas fo l lowsIn Fig. 1 let the transi t be at 0
,and A andB be the pegs . AO
i s a horizontal through A ,so that GB i s the difference ofelevat ion
FIG . 1 .
ofA and B . Suppose l ine of s ight to cut the rods at E and D,
we m ust find DG so that the target m ay be set at the proper read
PRELIM IN ARY SURVEYS. 13
ing to m ake the l ine of s ight horizontal . Let OF : a, FG
EA DE CB 16. Draw DE paral lel to CA and 0 G;then EH : 7'
+ k
From sim i lar triangles
a + bDG z EH b = (fr+ k
Set the target at a reading GB G1) s ight to G,and t i .
l ine of sight w i ll be horizontal . Bring the bubble to the center o fi t s run while the telescope is in thi s posit ion
,and the adjust .
m ent is com plete.
If des ired,a correct ion for the curvature of the earth and re
fract ion m ay b e introduced, but for shor t s ights th i s i s a uselessrefinem ent .
2 5. T he V ernier i s an auxi l iary scale for m easuring sm al lerdiv i sions than those graduated on the l im b . T here are two
c lasses , the di rec t -reading and the retrograde,according as the
frac tional parts of l im b readings are taken on that side of thezero of vernier scale towards wh ich the vernier has moved withrespec t to the l im b
,or the reverse. On the direct vernier a cer
tain num ber of div isions on the vernier equal s the same num berofdiv i s ions on the l i m b
,less one on the retrograde there i s one
m ore div i s ion on l im b than on vernier when the sam e space i scovered by both .
26. T he L east C ount of a vernier i s the sm al lest subdiv is ion of
l im b graduat ion that can be read by it,and equal s the difi erence
of one space on l im b and one on vernier .Let I value of one space on l im b
o value of one space on verniern 2 :num ber of spaces on vernier.
T hen for the di rect vernierne = (n — 1)l ;
from which we get the least count ,l — e
For the retrograde vernierno r:(n 1 )l,
14 A F IELD - MAN UAL FOR RAILROAD ENGIN EERS.
from wh ich the least count i s'v — l
the sam e resul t as found for the di rec t vernier .So
,t o find the least count D ivide the ealue of one lim b space by
the num ber of sp aces on the eernz’
er .
For exam ple If the l i m b of a transi t i s div ided to half-degreesand the num ber of spaces on the vernier i s 30
,the least countwi l l be di vided by 30, or 510 of a degree— that is , 1 m inute.
2 7 . T o Read a V ernier,take the num ber of the last div i s ion on
l i m b back of the vernier zero,then look along the vernier unt i l a
l ine i s found t o co inc ide w i th a l ine on the l i m b ; add the num berof th i s vernier l ine, m ul t ipl ied by the least count , t o the scalereading,
and the resul t w i l l be the required reading .
0 . Accessories .
T he Gradienter .
2 8 . T he G radienter consists of a tangent - screw hav ing a
m icro m eter-head, attached to one of the standards of the transi tand capable of being c lam ped to the hor i zontal axi s of the telescope . It i s used— as i ts nam e indicates— in running grades , andi t accurately m easures a sm al l ver tical angle in term s of i t s tangent . T he screw i s so cut that one revolut ion m oves the telescope through an angle whose tangent at one hundred feet fromthe ins trum ent has a certa in value,
usual ly one foot . T he gradua ted head i s div ided into 100 par ts, so that one divi sion corresponds to ft . at 100 feet from instrum ent .T o run a g iven gradient , b r ing the telescope level and read the
m i c rom eter -head of screw ; then turn the sc rew as m any div i s ionsas there are hundredth s of a foot r ise or fa l l in 100 feet , and w i th
1arget set at the height of the horizontal ax i s , po ints on the
surface corresponding to the given grade can be found.
For exam ple T o run a per cent grade,m ove the m icrom
eter m i lled head 75graduat ions from the hor i zontal .“'
hen used as a T e lem eter,we m ay ei ther m easure the space
on the rod m oved over by the l ine of s ight for a given num ber ofrevo lutions of the screw
,or we m ay note the num ber of revolu
t ions requi red to m ove the l ine of sight over a certa in space on
m e
tau“
PRELIM IN ARY SURVEYS.
T he Stadia , or Telem eteo'.
J
2 9 . T he S tadia Is an instrum ent for determ ining the distanceof a po int fro m the observer by no t ing the space intercepted on a
rod by a given v i sual angle,as deter m ined by two aux i l iary w i res
paral lel t o ,and equidistant from ,
the ho r i zontal w i re of the trans i ttelescope.
When used wi th an o rdinary level ing-rod the wi resshould be adjustable ; if they are fixed (which for som e reasonsi s preferable) , the rod m ust be graduated t o correspond. In
addi t ion to the distance o fa po int from the instrum ent,the differ
ence of elevation i s determ ined by observ ing the angle m ade byl ine of s ight wi th the horizontal when the m iddle horizontal w i recuts a po int on the rod as high above the ground as i s the cent reof the telescope.
T he horizontal pos it ion of the po int i s deter m ined from i tsmagnet ic hear ing, or the az im uth of l ine of s ight w i th referenceto som e fixed l ine,
usual ly the north -south l ine .
30 . L ine of Sight H or izontal . — In Fig. 2 let a and b be thestadia w i res , AB the intercept on the rod. T he secondary axes
FIG . 2 .
(LA and bB pass through the opt ical center 0 . Let h ab,
r AB. d di stance of cross -wires from objec tive,D di stance
of rod from ob jec tive.
From sim i lar tr iangles ,From optics ,
1 1 1
( l D f
in whichf i s the focal length of object ive.
El im inat ing d from these two equat ions,
16 A F I ELD -MANUAL FOR RAILROAD EN GI N EERS.
Let 0 be the m ean di stance of object ive from center of instruim ent". Adding th is to 1) gives , for the di stance of the rod fromthe center of the instrum ent
,
m ay be made constant , when (2) becomesl = a + kr.
3 1 . L ine of Sight Inc lined.— When the line of s ight i s not
level i t i s diffi cult to hold the rod perpendicular thereto ; hencethe rod i s held vert ical , the angle of inc l inat ion m easured, and a
correc t ion appl ied. In Fig. 3
let r CD be the reading on rod held vert ical7"
2 FE ,the reading perpendicular to l ine of sigh t
H : AG,the horizontal di stance from A t o B
V : BG , the difference of elevat ion between A and B ;
n BAG,the angle of inc l inat ion of l ine of sigh t .
Assum e angles A PB and AER from wh ich they rarelydifier m ore than 15’ to T hen, since FBC z
. n,
7"
z; 7'
cos n.
3 4 . Station N um b ers should begin w ith zero for the ini t ialstake
,and are m arked on rear side of stake,
from the top downward,the num ber of the prel im inary , A , B , 0 ,
et c . ,being-
m arkedonithe forward s ide. T he m arking should be w i th k iel , or c rayonthat w i l l w i thstand the act ion of sun and rain. Stakes m ay be
set every hundred feet or only at even stations,as preferred.
35. H ub s ,or P lugs , are transit turning-po ints . and are short ,
fiat -topped stakes dr iven into the ground flush w i th the surface.
T he flag i s held on the top and careful ly al igned,the pos ition of.
the po int being m arked by a tack . A spec ial tack w i th concavehead offers a foothold for point of flag when used in backs ighting.
! About 10 inches to the left of and with .num bered sidf
fac ing the hub i s driven a guard-stake t o m ark i ts posi tion.
36. Referenc e-
p oints are two or more hubs , w i th guard-stakesin each of two l ines m aking a good intersec tion angle at the
po int whose posi t ion they serve to locate. T hey should be driver.
beyond reach of di sturbance, and are used in replac ing a dis
located hub .
T hese need rarely be used on prel im inary.
3 7 . A l ignment .— It i s not intended that the prel im inary and
locat ion l ines occupy exact ly the sam e position ; hence considerable lati tude i s al lowab le in the size and num ber of anglesturned
,care being taken,
however,that the m axi m um curvature
need not be exceeded on locat ion. Large trees and other obstruct ions m ay be avo ided by turning a sm al l angle until the obstac lehas been passed,
then m ak ing a deflect ion in the opposite sense.
Bearings of tangent s are taken w i th the needle,to serve as a
check on the angle read on the plates .In easy country not requi ring a topographic party large angles
should no t be turned,a success ion of sm al l ones w i th short inter
vening tangents being subst ituted in o rder to m ake the prelim inary profi le approxim ate m ore c losely to the locat ion p rofile.
T hese short tangents m ay convenient ly be the long chords of thecurve that i s to fo l low .
Such a tack is m anufac tured by the A . S . A loe Co . , St . Louis.
PRELIN INARY SURVEYS.
3 8 . T he T ransit N o tes m ay be kept in the form below ,wh ich
shows both pages of the note-book . T he notes run from the
bottom up , the right-hand page being reserved for sketches the
red l ine up the middle of the page represents the transi t l ine,whether straight or broken,to which the sketches must be
adjusted.
Sta . Angle. nggfid '11
1311
53221;
Rem arks and Sketches.
68G
EQ 20° 0'L . N . 1° 48’ W. N . 1° 45’ W.
6656463 0 6° 2’R. N . 18° 12’ E . N. 18° 15
’ E .
6261
3 9 . Stadia M ethods for P relim inary Surv ey s.— P rel im inary '
l ines are usual ly run w i th the t rans i t, but the com pass w i l lanswer nearly as wel l in m ost cases , besides adm it ting of m orerapid work . T he transi t and stadia m ethod migh t wel l be em
ployed,and would effec t considerable saving in the cost of pre
l im inary s urveys . For som e reason rai l road engineers have notregarded it w i th favor
,though i t i s extensively em ployed in
topographic surveying where the m ap i s t o be used for work thatis often m ore prec i se than needed for rai l road prel im inaries .Part icularly is this m ethod appl icable to explorat ion l ines .W i th the transit and stadia the ent i re surveying corps need not
exceed five or s ix m en,the instrum ent m an act ing as t rans i tm an
,
leveler , and topographer all in one. T he only objection wouldseem to be in the am ount of reduct ion the notes would need ;
however, w i th tables and sl ide-rule (see 3 3 ) th is work m ay be
very rapidly done. For vertical angles of less than one degreethe horizontal reduct ion can be neglected,
and wi th side readingsfor topography the angle m ay be 5or 10 degrees without necessitat ing the correct ion. Vertical heights are found by the sl iderule or
'
by charts .T hi s m ethod would real ly necessitate the m aking of a topo
graphic m ap along a narrow str ip of country, from which theprofile could readi ly be taken. W ith a ski lled observer and twoto four rodm en the progress m ay ;be m ore rapid
,and ful ly as
good for the purpose intended as the m ore expens i ve m ethodusual ly em ployed.
20 A F IELD -MAN L’
AL FOR RA ILROAD EN GIN EERS.
T he t ransi t need only be set at al ternate stations (wh i ch m ay be
any length w i th in the reading l im its of the Wires) , the bearings too ther stat ions and po int s ofi the line be ing taken w i t h the need le.
T he hor i zontal angle should al so be read on the p lates for pointson stadia l ine
,as a check on the bear ings .
E. Obstacles in Tangent .
4 0 . Obstruct ions to vis ion and m easurem ent in tangent m ay be
avo ided in a num ber of ways , a few of wh i ch are given in the
fo l low ing problems . Other m ethods of avoiding t hem w il l suggest them selves in spec ial cases .T he sam e dev ices m ay be used on locat ion,
but i t i s m ore impo rtant to m a inta in a c lear sightway then so , when possible, we
should rem ove the obstruct ion.
4 1 . T o P as s an Ob stac le b y M eans of P arallel Lines .
— In
Fig. 4 , 0 i s the obstruct ion,AB the obstructed line. At B set
FIG . 4.
t rans i t turn 90 °
and measure BF long enough to c lear ob struc~
t ion.
Q
Set t rans i t at F,m ake BFG and m easure FG .
Move to G and backs ight to F ,m ak ing FGC Measure
GU: FB ,and m o ve t o C, where the angle GUI) i s made equal
to Cl ) i s the des ired l ine,and BU: FG.
Otherwise, at A and B erect perpendi culars ; take BF : AE ;
produce EF,and at G and H , beyond 0 ,
erect perpendiculars m ak
ing GU HI) FB . CD wi l l be the des i red line, andBU: F64 2 . T o P as s an Obs ta c le b y Angular D efiec tions.
GEN ERAL CASE . Angle anythm g less than
At B (Fig . 5) on the obstructed l ine deflect an angle a to one
side and m easure BC,taking 0 so that after deflect ing 2a t o the
other s ide CD w i l l c lear the obs t ruct ion. Make CD =BC and
deflect an angle a t o the sam e side as at B ; D E Wil l lie in ABproduced. Draw CH perpendicular to BD ; then
BD = BH+ HI ) z 28 0 cos a .
FIG . 5.
EXAMPLE .— Suppose a 14
°
BC CD 520 ft .
BD 2 X 520 x 2 feet .SPEC IAL CASE . A ngle60 degrees.
_
In th i s case the tr iangle BDF (Fig. 6) is equilateral and BF
BD DF
Should i t be inconvenient t o run to D we m ay stop at 0 ,having
m easured BC. At 0 deflect 60°
and measure CE at E again de
FIG . 6.
fiec t 60°
and make EF : BC'. At F a final deflect ion of60°
in the
opposite sense will put the telescope in the dwired l ine, PG, and
BF = BC+ CE . (5a )
4 3 . T o P ass an Obstruc tion,such as a River, when the Pre
c eding M ethods are Inapplic ab le.
Fm s'r CA SE . P oint beyond obatm tabn a
In Fig. 7 let BC be required.
FIG . 7 .
At B erect and m easure the perpendicular BD set inst rum entat D and m easure angle BBC : (1 then
BU: 3 1) t an a .
22 A F IELD -MANUAL FOR RAILROAD ENGINEERS.
Or,if a trigonom et ric tab le i s not at hand,
m ake ODE : 90°
and
fix the po int E where DE intersec ts AB ; m easur ing EB thereresul ts
,from sim i lar triangles
whence
Otherwise, if a right angle at B i s not convenient, measureangles CBD b, BDC
’a,and s ideBD . T hem e 180
°
From triangle BB 0 ,
BC B1)S111 6
EXAMPLE .~— a 0 B1) 400 feet.
sin 56°
By BO: 400sin 54
.feet.
SECOND CASE . P oint beyond obstruction invisible.
At B (Fig. 8 ) m easure angle b and l ine BE ; m ove to E and
measure angle y ,and set hubs on line EC so the l ine EUWi l l pass
FIG . 8 .
between them . Angle z z:ECB 180 (b y) . T hen from triangle BEG
BC: BE
Produce EB to D ,where D0 wi l l be sure to c lear obstruction;
PRELIM IN ARY SURVEYS.
T he sum and difference of a and {c are now known, so both m ay
be readi ly found.
At 1) set ohethe angle a w i th the transi t , and have the chainmen
stretch a cord between the hubs set on l ine EUat C’. Now signa l
the fiagm an to m ove h is rod along th i s cord unt i l the vertical w i recuts i t at 0 . Set a hub here and place the transi t over it . Sightto D or E ,
reverse telescope and deflect into OH .
ART ICLE 4 .—T HE LEVEL PARTY.
4 4 . T he L ev el Party consi sts general ly of two members, theleveler and a rodm an ; som et im es an axem an i s added to keep theiodm an suppl ied with pegs for turning-po ints and in c lear ing thel ine of sight for the level . As the party fo l lows the transi t l i ttleor no c learing wi l l be needed. T he instrum ents used are a level
,
a rod,and a hand-axe or hatchet .
4 5. T he L eveler m akes all necessary observat ions with h i sinstrum ent
,keeping a neat
,accurate record of readings and ele
vat ions also the pos i t ions and elevat ions of benches and turningpo ints . He should work out elevations of s tations wh ile the rod
m an i s going from one stat ion to the next he must see that therodm an gives him readings at points where the longitudinal slope '
changes suddenly , recording the plus . He m ust plo t h is profileatnight , or at such t i m es as the ch ief of party is l ikely to need it .The rodm an
’s readings at turning-po ints should be checked.
46. T he Rodm an holds h is rod at each station, cal l ing out the
number . Ifstakes are set only at even s tations , he must hold hisrod midway between stakes , the point being found by pacing thedi stance. T arget-readings need only be taken at tu rning-
pointsand benches , and the rodman should keep a record of these inhis “ peg-book ,
”checking the calculations of leveler for heights
of instrument and elevations (if turning-
points . At any‘
m arked
surface change he will hold his rod, cal l ing ont the plus to leveler.He must assist the leveler in plott ing up the notes.
4 . Adjustm ents of the Level.
4 7 . T o Adjust the L ine of C ollim ation i s to bring the intersec tion of the c ross-w i res into the op tical axis of the telescopeSet up and level the instrum ent . then bring the vertical wireinto coinc idence wi th a plumb - line or vertical edge of a building ,
24 A F I ELD -MANUAL FOR RAILROAD EN GIN EERS.
at the mean length of sight , and note if the vertical w ire i s t rulyparal lel thereto . If it is not , loosen the capstan-headed screwsho lding cross -wire ring and turn sl igh tl y so that the wire i sparal lel to the vertical line.
Loosen the wye-c l ips and bring the vertical wire into co in
cidence wi th the l ine and clamp the instrument . Rotate the
telescope in the wyes 180° and note if the wire coinc ides w ith thel ine. If no t , correc t one ha the
'
error by loosening one and
tightening the opposi te ,of the capstan-headed sc rews that hold
the c ross-wire ring in place,remembering that the image of
the cross wires is inverted by the eyepiece.
T urn the telescope unti l the horizontal wi re i s paral lel to the
plumb - l ine or edge of building, and make the same test andcorrection. Repeat for both wires . T he horizontal wi re i s theone on which the accu racy of level ing depends , but i t i s wise tohave both adj usted. T hei r intersec tion should remain on a pointduring a complete rotat ion of the telescope in the wyes .4 8 . T o Adjust the L ev el-b ub b le i s to bring the axis of the
level o tube into the same vertical planewith the l ine ofcol l imat ion,
and to make the bubble stand at the center when the l ine of sightis horizontal .Since the axis of the telescope co incides with the l ine joining
t he center of the wye-r ings (which requires these to be of the
same si ze) , i t i s sufficient to make the axis of the bubble paral lelto this l ine.
(a ) W i th the telescope over one diagonal pai r of level ingscrews and the cl ips loosened, bring the bubble to the center ofi ts run then turn the telescope, in thewyes , a l i ttle to ei ther sideof the vert ical plane through the telescope and note if the bubbleremains at the center . Ifnot , correct the error by means of thescrew at endof the level -tube case arranged for lateral m ovement .Repeat unti l the tube m ay be rotated half an inch or more to
ei ther side of vert ical w i thout movem ent of the bubble. T hisadj ustment is made m erely to prevent error from fai l ure to set
level - tube vertical ly beneath telescope.
(b) W i th the wye-cl ips opened wel l out , again bring the bubbleto the center of i ts run ; rem ove the telescope fro m wyes andturn it end for end,
then careful ly replace it in thewyes . Shouldthe bubble fai l to rem ain at the center , bring i t ha lfway back byrai sing t he lower o r depressing the high er end of tu be at the.
po ints of attachment to telescope. Relevel and repeat as a test .
26 A FI ELD -MANUAL FOR RAILROA D EN GINEERS.
sensibly from that obtained by dividing by 21? 0 . T hereforewe wri te
For t 1 mile R 3963 m i les, 0 abou t 8 inches . Hencefor any other d1stance in miles we have, for c,
e z 8 X t 2 inches .
The correct ion for refrac t ion is abou t a, hence we have,from
or, closely enough( 10)
EXAMPLE .
— Wh11t i s the correc t ion for a half-mile sight ?For one eighth of a mile ?
By c 8 X (39 2
”for first case,
and e 8 x (t )? z 0 . 125for second case.
By (10) the hual correction i sc x 2 z 1 . 7 for first case,
a z X for second case.
52 . T he D ifferenc e of E lev ation between two points not sofar apart but that a rod m ay be read on each from some intermediate po int m ay be readi ly found from these rod-readings .
In Fig. 10 let the instrument be at 1 ,A and B the points
whose difference of elevat ion i s desi red. Let r AD, 7" B 0 .
Since the l ine of sight , DO,i s horizontal , the difference of
FIG . 10.
elevat ion wil l evidently be fr’
fr . When the distance fromI to A equals that from I to B the errors due to curvatureevidently balance.
PRELIM INARY SURVEYS.When the points are so situated that the rod cannot be read
on both from one intermediate posi t ion of the instrument, an
FIG. 11.
auxil iary point or po ints must be used and readings taken on
t hese po ints in pai rs . T hus in Fig. 11 suppose the differenceof elevation ofA and B requi redW i th the inst rument '
at I read on A and some intermediatepoint E . Considering the backsights as plus and foresights as
minus, the difi
‘
erence of elevation ofA and E i s AD FE.
Again,with the instrum ent at I the difference of elevat ion ofE
andB i s GE 0B. T he sum of these differences equal s the difference of elevat ion ofA and B , and m ay be writ ten (AD GE)
(E17 44 GB) , or, in general , the sum of the bachez’
ghts less the sum
of theforesights equals thedz'
fl‘
erenee of eleea tz'
on.
C. Fie/d-work.
53 . A D a tum i s a level su rface so taken that it shal l lie belowthe lowest po int l ikely to be reached by the profile, to which thesurface elevations are referred. I t i s often spoken of as the
datum -l ine or datum -plane, and is the zero ofelevat ions .54 . A Bench- m ark i s a permanent mark ,
such as a copper orother bol t let into the top of a sol idly fixed stone, whose heightabove the datum is known; it m ay be simply a mark on a stone
,
or a tack driven into the projec ting roo t of a t ree,upon which
the rod m ay be read. In any case i t must be so si tuated that i tcannot change . its elevation nor i s likely to be di stu rbed wi thinthe time for which it is intended to be used as a standard ofreference.
T he elevation should be m arked on some object adjacent tothe bench , with the letters B. M. indicat ing the nature of the
po int .
28 A F IELD -MANUAL roe RA ILROAD ENGINEERS.
55. T he F ield-work consi sts in finding the elevat ion of a
number of points on the l ine establ ished by transit party suffi
c ient to give, when plotted, a fai rly correct outl ine of the surfaceas seen in profile.
A bench -m ark is taken at the beginning of the l ine, and its dis
tance above mean sea- level or other datum is known or assumed.
T he level is set wit h one pai r of level ing screws in the line to be
run (ih order that any change in the posi tion of the bubble m ay
be easily correc ted) , and the rod i s read on the bench . T his reading plus the elevat ion of bench gives the height of instrument(H . I . ) above the datum .
Readings are taken at every hundred feet along the line, or
oftener if the surface changes greatly, unti l a point i s reachedbeyond which it i s desi red to move the level . A peg i s drivenfirm ly into the ground and the rod read on th is ; the height ofinstrument less the rod reading wil l gi ve i ts elevation, as i t wi l lfor the. i11 term ediate points . T his point i s a tem porary .
benchand is cal led a t urning-po int . I t should be marked by ,
a guardstake if i t i s desi red to use it again. T he instrum ent i s now car
ried beyond the turning-point,set up , and the whole process
repeated. Benches and turningp oints should be read to hun
dredths or thousandths of a foo t , interm ediate points to tenths .T urning-points are marked Q or T . P . in the notes
,and thei r
posit ions , as al so the b ench -marks, noted by both leveler and
rodman in thei r note-books.56. T he L ev el N otes m ay be kept in any convenient formthat i s easily understood. T he fol lowing i s used more extensively ,
perhaps , than any other:F . S. E lev . Rem arks.
{B M. on root of L . O . tree 60’ tor ight of line.
B era the elevation of the datum was taken 200. 00 feet belowthe first bench -m ark . T he instrum ent was set up near Stat ion 2 ,
g
qc
oo
-qoo
co
00
e
m
9
<0
.
vuo
so
1Ou p eg at 4+30’ 20’ to left of l ine,
by sm al l P . O . tree.
trac ted from the yields an elevat ion of T he elevations of other points were determined in the same way . A l i ttlebeyond Stat ion 4 the rodm an drove a peg and held the rod on it ,
y ielding a reading of and an elevat ion of T he
instru m ent was then m oved to a po int near Station 7 and a reading of taken on the peg ; th is added to made thenew H. I . and the process continued wi th thi s H. I .
In most cases i t w i l l be suffi cient to read benches and turningpoints to hundredths and intermediate points to tenths .It will be seen from the notes that any error in ea turning-point
causes the same error in all succeeding po ints . T o guard againstthis the rodman i s requi red t o keep a pegb ook ,
”in which the
heights of instru m ent and elevat ions of turning-points are re
corded, and which must check wi th the leveler ’s record.
57 . Wind and sunsh ine affect the accuracy of the work withthe level , as i s also the case wi th the t ransi t . For very greataccuracy a cal m , cloudy day i s the best , but the rail road engineercannot always choose the best ti m es for his work , and must takesuch precautions as m ay be possible while he exercises the greatest care to prevent and detec t errors . T he adj ustments shouldbe tested at least once a week , even when the greatest care hasbeen taken, for unequal expansion and other causes m ay con
sp i re to cause them to change.
By making foresights and backsigh ts to turning-points abou tequal the error due to cu rvature w i l l be eliminated; the readingsof rodm an at these points should al so be checked. T he rodmanshould hold his red ver t ical
,which is som et imes accomplished
by m eans of a level attached to rod; or the leveler can tel l by hi svertical w i re when the rod is in the same vertical plane w i th theinstru m ent , and by causing the rodm an to wave hi s rod back andfo rth slow ly, after clam ping the target, he can tel l if the hor izoulai w i re j ust bisec ts the target at i ts highest posit ion.
58 . T he Rod should he graduated to feet and tenth s , readingby tartret at turning-points and benches ; interm ediate readingsare m ade by the leveler at his instrum ent . S t rength and dura
b ility are essential qual i ties . T he Philadel phia rod seem s to
30 A FIELD-MAN UAL FOR RAILROA D EN GIN EERS.
answer the purpose as wel l as any other now m anufac tured; theT roy rod m ay be used in the same manner as the Philadelph iarod, bu t i s l ighter and less able to stand rough usage.
ART ICLE 5. T HE T OPOGRAPH IC PARTY.
59 . T he T opograc P arty fol lows the level and secures all
the data necessary for making an accurate contour-m ap of a stripof country extending as far each side of the prel iminary as m ayb e needed for the intel ligent projection of t he locat ion- l ine.
T hi s distance m ay vary from 50 to 300 or 400 feet , i ts w idth depending on the diffi culties to be encountered and the degree of
prec is ion wi th which the prel iminary approximates to the final
locat ion-l ine. T he lateral slope of surface is obtained at the
stations of preliminary by m eans of the hand-level and tape, bythe slope-level or c l inometer, by cross sect ion rods
,or by the
transit and stadia. St ric t ly speak ing the topography incl udes allthe surface features , but for rai l road work the surface elevations,strea m s , and nature of surface are the most im portant y it m ay
be necessary to note the posit ions of roads , buildings, etc . , and
should always be done when prac ti cable w ithou t undue loss ofti m e. A p ocket eom pass will be of use in observing the bearings of l ines .
60 . T here are two m ethods of recording the data ob tained;one by means of notes and sketches in a book , the other bydraw ing the contours di rec tly on the field- sheet as the data are
obtained. Stat ion elevat ions can be taken di rect from the leveler’
s
notes , and consti tute the base on which the contour elevationsrest .Suppose the hand- level to be used and the no tes kept in a book ,
to be afterwards t ransferred to the m ap . Starting wi th the
known center elevation, the topographer notes the heigh t ofhiseye above the ground and cal culates the heigh t of center aboveor below the next contour ; from this the reading of the rod whenheld on th i s contour is found,
being the height of station abovecontour plus the height of eye. He direc ts the slopem an in or
out on a l ine at right angles to prel iminary unti l thi s reading isgiven by the hand-level ; the distance out is then measured and
recorded, j ust as in setting s10 pe-stakes , and the slopem an di
reeled into posi tion on the next contour , in the same m anner.
T hus if5-foo t contour- interval s are employed, and the station
PRELIM IN ARY SURV EYS
elevation is feet and the height of eye feet , we shal l havefor the reading at the 320-foot contourMotion the slopem an down the slope unti l hi s rod reads and
measure the distance out , suppose 21 feet . T he 315-foot contourwil l be 5 feet lower , giving a reading of Wh ich m ay be
found in l ike manner at , say , 80 feet out . As the rod reads onlyto abou t 12 feet the topographer must m ove out to this las t point ,and with the reading 5 find the 310-foot contou r inthe same way . Ou the uph i l l side the 325-foot contou r wil l befound wi th a reading of (325 feet, and othercontours in l ikemanner .T he notes m ay be wri tten thus
Left . Center E lev . Right.305 3 10 315 320
321 6325 330 335 340
80’
21 27'
56’
80’
1 12
T he number; above the l ine i s the contour elevat ion,the num
ber below i ts distance out from center .If preferred the elevation can be taken at regulardistanees out
and recorded as above:the .p osi tion of the contour wil l then befound by interpolat ion when mapping the work .
61 . If the topography is to be plotted in as the work progressesthe topographer must have a l ight drawing-board wi th a pocketand flap on back for holding the sheets on which the transit-l inehas been plot ted the night before ; the station elevat ions are
marked on the l ine and the contour posi tions spotted in as oh
tained by slopem en, after whi ch the contours are sketched in.
Po ints where contours cross transi t -l ine are found in the samemanner as side po ints . T he size of the sheets wil l depend on the
taste of topographer and si ze of drawing-board; 17x24 to 19x28i-aches are good sizes .T he topographer wil l soon learn to guess at the position his
contours wi l l occupy at the nex t stat ion ahead, and wil l sketchthem in l ightly , to be erased and
‘
correc ted when necessary . I t isoften sufiic ient to take lateral readings at every second or thirdstat ion.
62 . If the Slop e o lev el i s used, the inclinat ion of the su rface i sobtained; then by the use of a scale construc ted to show the
32 A El ELD—MAN UAL FOR ' RA I LROA D EN GI N EERS.
ho rizontal distance apar t of contours , for the given contour ihterval, for slopes varying fro m 1
°
to the posi tion of contourscan at once he spotted on the m ap . W
'
el l ington recommends theuse of the al t azi m u th as perm i tt ing the em ployment of ei thermethod at will— the al tazimuth being merely a haud- level w itha c l inometer a t tached.
63 . C ross - sec t ion Rods are measuring- rods 10 or 12 feet longcarrying a level -bubble. By plac ing one end at the center ,bringing the rod horizontal , and noting the height of the end of
rod on the down hil l s ide, the slope ll l’
tty t'
eadilybe obtained and
the contours worked in as before. For very rough ,broken
ground this m ethod m ay be preferable to either of the others .64 . If the T ransi t and Stadi a are em ployed,
very elaboratetopography m ay be taken with very li t tle field work ,
but the oh
servations requi re considerable reduct ion. W i th a su itable topographic protrac tor and the sl ide-rule m entioned in 3 3 , the large.nu m ber of points that m ay be obtained from each set ting of thetransi t m ay be readily plot ted and thei r elevations m arked on the
plot , after which the contour- l ines can be worked in,and other
features mapped. For smal l vert ical angles nohorizontal redue '
t ion is needed.While not general ly favo red by railroad engineers in the past ,this m ethod i s p robably the m os t rapid and economical of any so
far employed in topographic work .
ART ICLE 6. PRELIMINARY EST IMATES.
65. After co m pleting the field-work of the prel i m inary surveythe par ty is usual ly disbanded , only the transi t m an,
leveler , andtoliographer being retained to assist the chief of party to completethe m ap , profile, and esti m ate of cost .
66. T he M ap m ay b edrawn to any su i table scale, but less than400 feet to the inch is no t to be reco m m ended where i t m ust beused in pro jecting location. T he transi t - line is laid down first
and the to pography worked in afterwards fro m the field-m ap or
topographer’s no tes . If i t i s wanted on a continuous sheet , the
transi t- l ine m ust first be drawn on a succession of sm al l sheets ,which are
'
adtled as the plo t ting progresses, a new sheet beingsl ipped under the edge of the preceding and tacked down when
34 A F IELD-MANUAL FOR RAILROAD EN GIN EERS.
Engineering expenses and unforeseen outlays that are sure toarise should have a l iberal al lowance.
69 . T he Report of the chief of party should set forth the advantages and probable cost of each of the several l ines run
when there i s more than one. Ou this report frequently dependswhether or not the l ine i s to be located, and it should be clearand exhaust ive,
though plainly and concisely worded. The m ap
and profile form an integral par t of the report and show fromwhat data the estimates were derived.
CHAPTER III .
LOCATION
ART ICLE 7 . PROJECT IN G LOCAT ION .
'70 . After the prel iminary has beenmapped and the topography
worked ih , the engineerproceeds to m ake a paper location for hisguidance in the field. T he sol ution of the varied and co m plexproblems that confront him are more or less interdependentT he guiding princ iple, appl icable to all departments of engineering, that the best structure is tha t whichfor the least cost best an
steers the purposefor which it was intended, should cont rol , eventhough the resul ting struc ture be inferior , in po int of scient ific
design, to some o ther . T he best road as regards construc tion and
grades m ay be a fai lu re because of excessive first cost , whilethe cheapest construc tion w i l l entai l such heavy operat ing ex
penses that i t m ay be equal ly unprofi table. T he al ignment mus tb e as free from curves as possible,
while heavy grades are at the
sam e t ime exc l uded; these two requi rements conflict and m ustbe as wel l adjusted as possible. T he amount of earthwork , ofbridging and other structures must be kept down to the lowestl imits .
'71 . Starting at the sum m i t of the most diffi cult port ion of the
route. assu m e a starting-po int and elevat ion; with the dividers setat such a distance to the scale of the m ap as will give a fal l of onecontour -space— or half space— for the assum ed grade,
step down'
the slope in such a way that the div iders fal l each time on the
next lower contour. o r half-space, according to the fal l assumed inset ting dividers . If curve compensat ion i s al lowed, the di vidersmust be reset for each curve, for the same fal l , since the gradew il l be slackenetl on curves . T he po ints at which the dividers
‘
fal l are l igh t ly spo t ted on the m ap and connec ted by a grade
c ont our , which represents the surface- l ine having the requi redgradient . T his l ine wil l b e too broken t o be used as a location
36 A F I ELD -MAN UAL FOR RAILROAD EN GIN EERS.
l ine, so we have then to draw on the m ap a succession of curvesand tangents that Wil l approx i m ate suffi cient ly close to it , at thesam e t i m e that a proper balance i s m aintained between earthworkand curvature.
Hav ing l ightly plotted the proposed l ine, the elevations aretransferred to profile
-paper , thus giv ing a profile of the l ine.W i th a fine thread stretched along the p rofi le, to represent thegrade- l ine, adjust the cuts and fills to sui t the nature of the work .
In general , fi lls are cheaper - than cuts bo th in construc t ion and
maintenance; and espec iall y is this true where a shallow surfacelayer of earth i s underlaid by rock . I t m ay happen that them aterial from excavat ion must be used in embankment , whenthe cuts and fi lls m ust be made to balance by sh ifting the gradel ine unti l this appears to be the case on the profile.
At the stream crossings the grade- l ine must be kept safelyabove high -water mark , so that suffi cient waterway i s provided,
and al lowance made therefor.After locat ing the m ost diffi cult portions pass on to the easierwork , returning later on . to study the effec t th is w il l have on the
part first located. It m ay be necessary to go over the projectionseveral t i m es before you can be reasonab ly su re that the bes t locat ion has been pro jec ted; even then the study of the l ine inthe
field will cause many of the detai ls to be al tered,someti m es
materially .
Long grades are to be preferred to shor t ones , bu t questions ofeconom y m ay necessi tate the latter in order to l ighten work °
carem us t be taken that the grades are not so badly chopped thatthey interfere with the easy riding of the train.
In projec ting the l ine i t w il l general ly be best to s trike the
curves first and draw the tangents afterwards , though i t somet i m es happens that long tangents w i l l contro l the curves ; whenthi s is the case the tangents are drawn to intersection and the
c urves afterwards put in.When t ransi t ion-curves are employed. a sl igh t ofi'
set should bemade at the beginning and end of curves to al low for thei r insertion in the field. T hese offsets w il l be so smal l that i t is uselessto attempt to show them to scale.
'72 . A Curv e-pro trac tor wil l be of m aterial assistance in find
ing the degree of curve requi red to uni te two tangents that havebeen laid down on the m ap . I t consi sts of a transparent , semic ircular protractor having a ser ies of enwes from 30
’ up to 8°
LOCAT ION .
plainly cu t upon it . T he cu rves are on both sides,those on the
reverse side having thei r concavi ties turned in an opposi te sensefro m those on the face. T he scale i s usual ly 400 feet to the inch ,and in any case the m ap and prot rac tor m ust be drawn to the
same scale. Sometimes a set of cardboard or hard-rubber curvesare used,
but they are inferior to the curve-protractor. T o use
it,simply prolong tangents to intersec tion and then place the
protractor so that the curve admitting of the best grade i s tangent to the two straight l ines . Mark the points of tangency ,
which w i l l be the beginning and end of curve. When the curvei s requi red to pass through a given point the proper curve m ay
be i m mediately found by trial , whereas the calcu lat ions wouldrequire som e l i ttle ti m e.
should never be al lowed on main lines . Suffi
c ient tangent should be interposed to al low space for easingo
the superelevation of outsid rai ls , or for the insertion of transi tiou -curves when these are t be employed.
7 3 . T he F ield C orp s is substantial ly that required on the pre
l iminary survey , and the methods ofwork pretty much the same,except that curves must now be run m and this necessi tates moreclearing. Iffi t st and second location ltnes are to be run (and i t i sreal economy to run both ) , i t wil l not’ be necessary to have the
stationing continuous on the fi rst, so the pluses ari sing frombacking up
”need only be noted and el iminated when the final
location- l ine i s r ou-curves are to be inserted, theyneed not be run the fi the proper offset being made at
t he P . T . or P . 0 . o f the c i rcular curves, which latter are t o be run.
On the final locat ion- l ine the stat ioning must be continuous,beginning wi th zero. T he stakes are m arked as on the pre
l i m inary su rvey , and all hubs that are l ikely to be used againmust be referenced in, the reference hubs being set wel l out ofthe way ofdisturbance by the plow or scraper.T he levelet should make bench m at ks evet y 1000 0 1 2000 feet ,
to be used 111 running check level s and in giving gt ades later ou.
From the paper locat ion the notes should be m ade up in theothee, to serve as a guide in the field; however , no at temp t shouldbe made to adhere rigidly to them
,since sl igh t errors in the
mapp ing wil l afi ec t the pro jected line, . w11ile in the field the l inem ay be sh ifted here and there so as to fi t the ground more snugly1nd accord more c losely w ith what the nature of the earthworkdemands .
38 A FI ELD -MANUAL FOR RAILROAD EN GINEERS.
T he highest skil l of the engineer i s requi red to secu re the bestlocation-l ine, and he should have al l the t i m e he needs. Unduehaste on location— as on reconnoissance and prel iminary— isalmost sure to resul t in increased cost of construct ion.
ART ICLE 8 . SIMPLE CURVES.
4 . Definitions andForm ulas.
7 4 . T he C ircular C urv es that are usually employed to uni testraigh t reaches of the rai l road m ay be simple,
‘ compound, or re
versed. T he use of reversed curves should, however , be l imi tedto turnouts and cross -overs .a . A Sim p le C urv e i s the arc of a circle.
b. A'
C om pound C urv e consists of two simple curves, ofdifferent radi i , both on the same side of a comm on tangent .c . A Rev ersed C urv e is made up of two curves of contrary
flexure having the same or different radi i , and a. common tangent .d. T he P oint of C urv e (P . is the end of tangent and begin
ning of curve, as at A , Fig. 12 .
FIG. 12.
e. T hei
P oint of T angent i s the end of curve and be
ginning of tangent , as at B ofFig. 12 .
f. T he P oint of Int ersec t ion is the po int where thetangent at the R C. and R T . intersect when produced. (D of
Fig.
g. T he Intersec tion Angle (D i s the angle at the PI . between the tangents meeting there, and equal s the angle at the
center .It . T he T angent D ist anc e ( T ) i s the length of the producedtangent measu red from the P . 0 . or P . T . to the R I . T he term
LOCAT ION . 39
tangent i s appl ied to any st raigh t portion of the l ine,bu t the letter
T will be used to designate the produced portion only .
T he M id-ordinate (M ) i s the portion of the radi us intercep ted between the arc and chord when i t cuts the chord at i tsmiddle point.j . T he E x ternal (E ) is the part of the radi us p roduced to theP . I . , intercepted between curve and the R I .
h. T he L ong C hord i s the chord joining the R C. and
R T . Frequently the term is appl ied to any chord longer thanthe uni t chord.
l. T he Radius wil l be denoted by R.
m . T he P oint of C om pound C urv e i s the point ofcommon tangency of the two branches of a compound curve.
(See Fig.
P .C .C . P
FIG . 13.
n. T he P oint of Rev ersed C urv e is the point of
common tangency of the two branches ofa reversed curve.
0 . T he D egree of C urv e (D ) i s the angle at the center subtended by the uni t chord. In the Uni ted States thi s chord i s 100feet , in England 66 feet , and where the metric system is employed i t i s taken at 20 meters . Any convenient chord lengthm ay be taken, but for uniformity American engineers haveadopted the chord of 100 feet, and unless otherwise stated it isalways so understood when we speak of the degree of curve.
Half the degree of curve i s cal led the defiec t ion-angle, sinceit i s the angle to be deflected from the tangent to the chord.
If there were any pract i cal method of measuring around the
curve instead of along the chord,an accurate and convenientrat io for expressing the radi us in term s of the degree would be
had. T hus ifD i s the angle at the center subtended by the a rc
of uni t length , we have, where a is this uni t are,
40 A FI ELD -MANUAL FOR RAI LROA D EN GIN EER
27zR= a
Hence
When a equal s 100 ft . this becomes
R varies inversely as D,so that knowing the radi us for a
curve, we should have only to divide thi s by D to get the radiusfor a D
°
curve.
Since the chord i s em ployed instead of the are, we determineR by means of the fol lowing problem
'75. G iv en the C hord O', and D egree ofC urv e D
,to F ind the
Radius R.
In Fig. 14 , AB i s the chord 0 , OE a perpendicular from the
center upon AR
FIG. 14.
From the right triangle AEOwe haveR sin 5D 2
R % 0 1 0 1Whencesin 2
1Dg cosec ED.
When Ois 100 ft
50 cosec 1D.
sm $ 1)15
42 A FIELD -MANUAL FOR RAILROAD EN GIN EERS.
76. The L ength ofC urv e (L ) is found by div iding the angleat the center (which equal s the intersec t ion angle) by the degreeof curve, the resul t being in chains and dec imal s of a chain. T he
number of P . 0 . L w i l l give the station number ofP . T .
EXAMP LE — T he P . 0 . ofa 4°
curve having I : 96°
30’ i s at sta.
104 Find L and the number of the R T . Here
L4
chains .
hence the number of P. T . i s
7 7 . U se of the T ab le ofFunc tions ofa One-degree C urv e.
In the locat ion of rai lway curves geometrical accuracy w il lfrequently be of less importance than rapidi ty of field work ,
so
long as errors are kept within cer tain l i m i ts .Ou tangents sl ight errors of al ignment m ay readi ly be detectedby the unaided eye, but on curves these are no t so apparent .Moreover i t is not l ikely that the trackmen wil l keep them up in
the exac t position of their location.
T o simpl ify and shorten the field computat ions engineers makeuse of a table of funct ions of a 1
°
curve , and assume these funct ions for other curves to vary inversely as thei r degree, or di rec tlyas thei r radi i . T able IX gi ves values of the tangent distances ,long chords , m id-ordinates, and external s for a 1
°
curve, the
radius ofwhich is taken as 5730 feet . T o find these func tionsfor other curves , div ide the tabular values by the degree of curve.
T he error result ing from this assum ption wil l , in any practicalcase, amount to no more than a few tenths or hundredths of a
foot .T able IX m ay also be used as a metric curve table, the tabularval ues being taken as meters instead offeet . If the uni t metricchord is 20 meters long, th is m ay be taken as one fifth of thetabular uni t chord; so to use the table m ul tiply the metric degreeby 5and enter the table with the result as a val ue ofD .
For instance, a 2° metric curve having 1 40° would have a
m id-ordinate equal to 3456 2 meters .
For the approx i m ate radi us of a m etric curve divide 5730 by 55730
t imes the degree. T hus a 4 ° metric curve would have R4 X5
fillifit" :
fl
LOCAT ION . 4 0
meters . For the exac t radi us make use of formula10
T hus for a 4°
curve having 20-meter chords Rsin 2
0
meters , a difference ofonly .04 meters .If a metri c curve i s to be retraced wi th a 100-ft . chain, we
convert the metri c degree to the degree referred to lo0 ~ ft . chordsby the relat ion that a 100-ft . chain chains of 20 meterseach ; a 20-m eter chain ft one foot meters ;one meter ft .
I t will som etimes be a suifi ciently close approximat ion to takethe 20 meter chain as two th i rds of a 100-ft . chain ; this wil l makethe metr i c curve nearly two thi rds of the degree the same curvewould have when laid out wi th a 100-ft . chain, and the curvewith100-ft . chords nearly three halves of the degree as laid out wi ththe 20-meter chain. T hus a 4 ° metric cu rve would be equ ivalentto a 6
°
curve laid out with a 100-ft . chain.
In the problems that fol low two m ethods of solu tion wil l begi ven when prac t icable— the first being rigid, while the second.is based on the use of T able IX . T o shorten the formulas thesubscript 1 wil l be,
written after the letters T , L . O. , M, and Ewhen these are the func tions ofa 1 ° curve. T hus T ,Zfi 28
° meanst he tangent distance for a 1
° curve when 1 : 7; 16°
the long chord for a 1 ° curve when I etc .
7 8 . T ables ofN atural and L ogarithm ic C ircular Func tions.
Many engineers prefer to work al together by tables of naturalsines , cosines . etc . , and t ime m ay often be saved by thei r use.
Nevertheless logari thmic tables are offrequent advantage, even inthe field
, and t he m ore im portant ones , such as the logarith m i csines , cosines , tangents , and cotangents, together wi th the logarithm s of numbers , are gi ven in the back of the book along withthe tables ofnatural functions.7 9 . G iv en R and C to F ind D .
From equat ion
8 0 . G iv en I and R (or D ) t o F ind T .
If 1) i s gi ven,find R by then in Fig. 15 from triangle
GAB we get
44 A FI ELD -MANUAL FOR RAILROAD ENGIN EERS.
'
BY T ABLE IX .
— Fiud the tabular val ue of T the givenangle I ; then
FIG . 15.
EXAMPLE.
— I 35°
D required T .
By T : t an 17°
50’
feet.By ( 14a ) , T feet
, a resul t differing from the
value found by the rigid method by only foot .8 1 . G iv en I and T to F ind R or D
FromT COt 1 1:o o o 0
tan g]15
T hen by T able I the degree m ay be found.
BY T ABLE IX.
8 2 . G iven I and D t o F ind the L ong C hord L O.
First find R by ( 12) or or by Table I ; then from thet riangle OAF ofFig. 15,
AR : R sin %1AG HAF : L . 0 . 2R sin (16)
LOCAT ION .
BY T ABLE IX .
—Find the tabular L O. for the given angle 1 ;then( 16a )
8 3 . G iv en the Radius R and any C hord (7 to Find t he
Ordinate t o the C urv e at any P oint .
FIRST MET HOD .
— In Fig. 16 let HE be the chord 0 ; B K : a
and K E b,the segments into which i t is di vided by the ordi
FIG . 16.
nate y . D raw the radius through K ; call the portion betweencho rd and curve y'. By geometry,
(2B y’
)y'
ab,
from which
2R —y"
But y' i s smal l compared with 218, and hence we write
N ow 3/ does no t diiIer sensibly from y in the cases m et with inprac t ice, so we wu‘ te
46 A F I ELD -MANUAL FOR RA ILROAD EN GIN EERS.
Ifwe writeR z5
750,formula becomes
2 X 573OH
LetE0
m,
100z n
,and subst l tute ln gtvm g
10000y
'm nD 0 .8 73m nD ,
or very nearly31 z gm nD .
y i s given in feet when m and n are in chains and dec imal s ora chain.
A t the mid point F ,m n
, and y M.
M z gni’D ,
CAUT ION .
— Fortunlas (17) and while very convenient forfield use in passing obstruc tions , are l iable to error when verylong chords or large values of D are used
,since they give resul ts
that are too smal l .Ifwe write the arcs HN , NE for a and b
,we shal l get resul ts
that are too large, yet about as near the true values as by taking772 and n to b e the segm ents o f the cho rd. T o i l lustrate we w i l lfind a few values of Ill and compare w i th the t rue values takenfro m T able V
0 0 0 0 0 0 0
From this i t appears we m ay use form u la (18 )— ai td ( 17 ) as
wel l— tak ing ei ther the segm ents of the are or chord fo r curvesnot exceeding 4 ° w i th ares up to 600 ft . ; for curves from 4
°
to 6
LOCAT ION .
they m ay be used up to 500 -t i . arcs, while for curves between6
°
and 8°
no t m o re than 400 feet of arc m ay be taken.SECOND METHon.
- First determine the m ido ordinate. In
triangle OEF,
OF : VRi — i fp ;
M Z FG Z R — VRi — i ci
.
T o find ordinate A C distant d from the m id-po int of EH, drawOB d paral lel to BE ; draw AR at right angles to HE. T hen
BA VB" d’ .
T herefore
CA y 2 VR? d e VR‘
(20)
T HIRD ME THOD .
— If the cho rd C is short , we m ay regard theare as an arc of a parabola,
for which i t i s known that ordinates vary as the produc t of the segm ents into which they dividethe chord. T he m id- o rdinate being known,
we have
From form u1a ( b) we have for y M a b 111 0 ,
T he m id-ordinate for any o ther chord C’ is
Hence
If 0'
10 , this gi vesM1 %M o o o o o
48 A FI ELn-MANUAL FOR RAILROAD EN G IN EERS.
T his last relat ion afi ords an easy m ethod of stak ing out a curvewhen the m id-ordinate of a given chord has been determ ined.
Firs t erec t the ordinate M at the mid point of the chord; thenj oin the ends of chord with the extremity of the ordinate j ustmeasu red; the lengths of these chords do not differ m uch from715 0 ; at their m id-points erect ordinates equal to %M, giving po intson the curve. Proceed in l ike m anner for other points unti l asuffi c ient number have been located.
8 4 . G iv en R and I t o F ind the“E x ternal E.
In Fig. 17 E : GE : OB 0 G.
But OB : R sec 4] and 0 G R.
E :R(sec % l (24)
BY T ABLE IX .
—Fiud E for a 1° curve for an intersect ion
angle I then
8 5. G iv en T and I t o F ind E .
In Fig. 17 draw BC perpendicu lar to AB, and produce AG to
FIG . 17 .
intersec t BOht C. BC is parallel to AO, and the triangles AGOand GBC
’are similar ; hence BC : BG E . In the right tr iangle
ABC,angle BAG: éBAF z
1 I . T herefore
E Z T taD %I . . o o o o oEXERCISE .
— Bel'i ve equat ion (25) from
50 A FIELD -MANUAL FOR RA I LROAD EN GINEERS.
When 0 z
d 200 sin5D .
1 0Ifwe wr1te sm gD
Rfrom (12)
'
1n formula there results
For curves up to 0 hence
10000
For curves from 7 ° to 0 therefore
ForR wri te 5JD30
’ and for C 100 , becomes
10000d
57301 .745D
and for 0 z 50. (30 becomes
2500d :
5—
730D _ .4363D 873
EXAMPLE .
— Find d for a 6° curve, 0 100 feet .By d 200 x feet .
0
By d feet .By d x 0 feet .9 0 . G iv en the C hord 0 and D egree of Curve D to F ind the
T angential D eflec t ion Offset t.
In Fig. 18 m ake EF (tangent at E ) equal to EA , and join Fwith A . D 1 aw EG to the m id-point of FA . Angle AEGGER :“D hence, i t om the figur ,e
LOCAT ION .
When 0 100 feet,t z :200 sin 4D .
Since 4D i s smal l,we m ay wr i te, without m aterial error ,
sin 4D 5sin %D ; then,writing sin 5D as in 8 9 , we get
Making 0 100 11. and writing R 123-
0gives
D 0 .873D .
When 0 50 feet , (33) yieldst z:
.
2 18D z. 436x
EXAMPLE .
— Find t for a 6° c urve, 0 z 100 ft .
By t 200 sin 1°
30'
ft .
'
By (33'
. 873 X 6 ft .
9 1 . T o F ind the Sub tangential D eflec t ion Offset t'
for a
Sub chord C"
FIRST MET HOD .— By form ula ( 13) find the angle at the center
subtended by the subchord ( J' ; cal l th is angle D'
. From
t'
2 0'
Si! ) ;}D (34)
SECOND MET HOD .
— In Fig. 19, with E as center str ike the arcsPG and AH , taking EF C
”and
EA 0 ; prolong EG to B . Now
assu m ing that the chords 0 'and 0
are propo rtional to their central.angles we have
From the similar sectors EFG FIG. 19.
and EAR, since EB C,
52 A F IELD-WIAN UAL FOR RAILROA D EN GIN EERS.
Mul tiply ing (a) and (6) together , term by term
Whence
EXAMPLE .
— Find t' for a 7 ° curve when 0 z 60 ft .
Here D’ x 7
°
(very nearly) 4°
By t'
2 x 60 x ft .
By z z ft .
By z' x ft .
9 2 . T o F ind the T angent Offset 2 .
In Fig. 20, EB : z i s the requi red ofi set . Let AE : n chainsl 00n feet . AE : FB
, the half-chordhaving the m id-ordinate AF : EB ;
hence we have, by formulaz gu
‘zl ) . (36)
In this formula we m ay take 71 to
be either the length of AE or the arc
AR,in chains . If taken equal to AE
the ofl’
sets will b e slightly too s m al l ,while if taken equal to AR they w i l lbe a l i ttle too large. T he use of the
formula is l im i ted to smal l values ofn and D
,as was po inted out in 8 3 .
(See CAUT ION . )
Formula (36) is easy of appl ication and of frequent use inlocat ing curves by offsets from the tangents . For curves up to4
°
71. m ay be as great as 3 , but for sharper curves it shouldbe less .EXAMPLE .
—Find six offsets to a 4°
curve at points 50 ft . apart ,measured around the curve.
FIG . 20.
LOCAT ION .
'
By successive appl ications of (36) we havez z gx X 4 : 0 .88 feetz :
n g, § _ X 4 :
Z : % X 4 X 4 = 14 00
z : % x - X 4 = 21 . 88
T he last value of z is in error by about ft , but for sett ingstakes on
“construc t ion th is difference i s no t material so long as
the al ignment beyond this po int does no t depend on it . In
setting track -centers the completed road- bed i s avai lable and the
stakes m ay be set wi th the transi t , in the usual way .
9 3 . D ifi'
erence in L ength of a C ircular Are and its L ong
C hord.
FIRST MET HOD .
—Let the central angle be a degrees . By
o0
3m 1a. 2R
Changing degrees to circular measure, a ( ia 7:meas . )
O
T he length of arc i s Eh R5? 3 . T hen
Arc chord Ra
c .
'
SECOND METHOD .
— Ah easy approximat ion m ay be found as
fol lows:
Referring to Fig. 17 , AE : 0, OF : 211 . Let AG b
From the righ t triangle AFG
From which
54 A F IELD-MANUAL FOR RAILROAD EN G IN EERS.
N eglecting the a:in denominator as smal l compared with 0
gives
T hen will 26 c 2 :
From Huygens’ approximat ion to the length of a circular are86 0
( see VVilliam sou’
s D ifferential Calculus, p . are
3
T herefore
Arc chord c g(2b c) . (e)
Insert ing the val ue of 2b c from (38) gi ves'
Arc chordWhen the arc i s not very great we m ay wri te 0 l 00m where
n. is the nu m ber of chains contained in the arc AE . Fromremembering that n. 212,
M I
Insert ing these values ofe and M in (d),
13 2
100m 800m D neat ly . (39)Are chord g
EXAMPLE .— Find the difference in length of arc and chord of
0. 4°
curve when n, 6stat ions .T he central angle is 4 x 6 then, from T able IV ,
C
By
Are chord X ft .
By
Are chord —6X 6X 6X 4 X 4 — 4 .32 ft
REMARK .
— F0 1'm ula (38 ) i s interest ing as showing what a co m
LOCATION .
paratively smal l increase in length of l ine is caused by a consitl
et able lateral deflec tion in al ignment . For instance, a lateraldeflec tion of 2000 feet is made at the mid point of a l inefeet long what w i l l be the increase in length ?
2(2000)Q
By (38 ) the 1ncrease 13 200 feet , giving for the
increased length feet .8 . Locating Sim ple Curves.
954 . T o L ocate a C urve with t he C hain by Offsets fromC hords P roduc ed.
In Fig. 21 let the P . 0 . fal l at B. IfB0 is a ful l chain, prolong
FIG . 21 .
the tangent AR to H,m ak ingEH EU EUwill equal t, which
m ay be calculated by or With B as center , strike an
arc with radius EH , and with H as center and t as radius strikean are . at 0 , where these arcs intersect, set a stake. P roduceBC to K ,
m ak ing OR BC CD ; strike the arc K D from C as
center ; make the chord K D z d, cal culated from or
Set a stake at D and proceed in l ike manner for the o therpoints unti l the P . T. i s reached,
'
where FP i s m ade equal to t.Usual ly the R C. does not fall at a full stat ion then RC t
’
,
which m ay be found by (34 ) or Using this value of t’
, we
locate 0 as above. At B make DR t'
, and prolong R0 to
L ; make LD t and set a stake at D . EM will equal d, and
m ay be located as before .
We m ay regard K D as equal to K L t, and, finding, XL ,
56 A F IELD -MANUAL FOR'
RAILROAD EN GIN EERS.
measure K D and set D without locat ing R. T o do this we havethe similar t riangles BBC and CK L,
from which
and therefore,since X C CD ,
XL t
In l ike manner at Fwe haveEF
PN _ tED
, and FP : t ,
henceNF : PN + t 1
’
Make EQ prolong QF, and we have the tangent at F.
EXAMPLE .
—Gi ven the R C. of a 5° curve at 106 20 and the
angle of intersection to locate the curve.
Here L?
53 stat ions.
T herefore the number of the P . T . i s
sta. 1 10 60.
BC in this case i s 80 ft . ,and by
t X 5 ft .
By z'
x a.
Set offH 0 ft . , and at D makeED x E
8
0
0
0ft .
At E make ME d by T h i s w il l be at sta. 109
at 1 10 set a stake by offsett ing ft . The last chord is 60 longand hence the offset
/60
Make EQ ft and p ro long QF , the terminal tangent .
NF : x —l x z: ft .
58 A F IELD-MAN UAL FOR RAI LROAD EN GIN EERS.
EXAMPLE .
— Locate three stations of a 4 °
curve by ofi sets every50 ft . on curse.
Referring to T able V ,the required offsets are
and By Tab le IV the distances measu redalong tangent are andWi th these val ues we can set out the curve ei ther way from A .
Had we used formula (36) we should have had for the valuesof the ofi sets and
96. T o L oc ate a C urv e b y Ofiset s from a giv en L ong
C hord.
F IG . 23 .
Let FK , Fig. 23 , be the given chord. We m ay compute theoffsets y . y, M by the methods of 8 3— o i which formula
y gm nD ,
i s the mos t convenient , with in the l imi ts of i ts appl icabil i tyand setting off these ordinates , locate the curve.
Or we m ay set 0 11 the m id-ordinate M : R v ers FOA at A
and at 0 set 0 11 y ? M R vers D ,making
AC z fl L s in D .
GE wil l be
y. M R vers 2D ,and AE R sin 2D .
AN OTHER MET HOD i s to find the angle K OF at the center, and
by T able IX determ ine BA 111 ; then by T ables V and IV
LOCAT ION .
determine BL, BN , LII , and 1 70 . T hen HO M BL , whichset off at C,
and other points in l ike manner .EXAMPLE .
— G ive11 the R C. of a 4° curve at stat ion 160 75,
the angle between tangent and chord requi red the offsetsnecessary to locate the curve.
I z:2 X 9 180
.
L1
4
8 stat ions .Hence the R T . fall s at sta. 165 25. T he
m id-point on c urve B fal ls at sta. 163 . By T able IX
70 5417 .64 ft .
By T able V the m id-ordinate for two stat ions of a 4° curve 18
BL
Hence H0
By T able IV . HL AO ft .
Measure A 0 ft . , and set off OH : ft . , and drive astake at H; In l ike manner find
GE : and A E : ft .
T he points P and Q are also located by means of the coordi
nates just determined.
If B had fal len at an odd station,the curve could have been
located in the same manner , H and P being 100 ft . from B, G'and
Q200, etc .
9 7 . T o L ocat e a Curv e With T ransit and C hain when the
D egree D or Radi us R is K nown.
If R i s given, determine D by then, since the angle inthe c i rcumference of a c i rcle is half the angle at the center subtended by the sam e chord, we m ay locate points on the curve bysuccessive deflec tions from the tangent .In Fig . 24 let the P . 0 . be at A , at which point set the transi t,
and wi th the vernier -plates c lamped at zero place the telescopein tangent either by sight ing the B ]. or by backsighting to somepoint in the tangent Defiect from the tangent half the angle atthe center for the sub-cho rd or chord, and direc t the head chainm an into l ine whi le the rear chainman holds h is end of the chain
60 A FIELD-MANUAL FOR RAILROAD EN G IN EERS.
at the transi t , the chain being kept taut . T he stakem an drives astake at the point where the head chainm an
’s flag rested, and the
rear chainm an advances to th is point . Deflect 5D from the chordAB j ust run, and while the rear chainman holds h is end of the
chain at B di rec t the head chainman into l ine at 0 . O ther pointsare located by deflec t ing an addi tional «
.e for each chord lengthmeasured, unti l a point E is reached to which i t i s desi rable to
F 1G . 24 .
move the transi t . T he angle FAE should not exceed abou tMove the transi t to E , backsight to A ,
and deflec t FEA EAR,
when the telesc0 pe w i l l be in tangent , and the curve can be continued unti l i t i s again necessary to move the t ransi t . At the
R T . pu t the telescope in tangent by backsight ing to the pointlast occupied by transi t and deflec ting the tangent ial angle as at
E . T he l ine m ay now be continued.
9 8 . T he Index -angle is read on the vernier -plate, and i s theangle between the tangent to the curve at the P . C. and any otherl ine passing through a po int on the curve when the telescope isdi rec ted along this l ine. I t i s most frequently taken as the anglebetween the ini tial and any subsequent tangent to the curve.
T hus at E the index -angleequal s EFP 2FAE . A t any pointon the curve the index -reading in tangent m ay be found by thefol lowing rule, which m ay be easi ly deduced from a figure:From double the index-angle tha tfixed thepoint subtract the index
( 171gle t'
u tangent a t the last point; the rem ainder is the index-angle
required.
9 9 . Subdeflec tion-angles m ay be found by 13) rigidly , or
approximately (and with sufficient accuracy except whenD i s verylarge) by assuming the central angles to be proportional to thei rcho rds . T hus on a 4
°
curve the cent ral angle for’
a sub -chord of
25ft . would be and the subdeflec t inn-angle
LOCAT ION .
EXAMPLE . : Loca1e a 4° curve to left when the P . C. i s at sta .
8 1 25and I : 32°
chains .
Hence the R T . wil l fa il a. sta. T he
first sub-chord i s 75ft . long, and the first deflec tion-angle wil l befound by
SID %6 I
—
216 1
°
By the approximate rule,‘ since 4D
whence 46 2 X 4 1°
30'
as before.W i th transi t at R C. deflect 1°
30'
from tangent , measure 75
feet , and set sta. 82. T hen a deflec tion of 3°
30' wil l determ ine
83 , 5°
30’
sta . 84 , 7°
30'
sta. 85. N ow rem ove transit to 85, andw ith - vernier at 7 ° 30' backsigh t to 8 1 25. Reverse telescopeand set vernier at 15° when the telescope will be in tangent .Ah index angle of 1 7
° wil l fix 86, and so ou.
T he last cho rd will be only 40 feet long,for which the sub
defiec tion-angle is T4505 of that is, T he index -angle fixing
the R T . is therefore 23 °
T o get in tangent at 89 40 backsight to sta. 85, wi th vernierat 23
°
48' then by the rule of 9 8 the index -reading i s (23° X
2 15°
32°
36’
I . Set the vernier a t th is reading and run
tangent .CAUT ION - It i s not good pract ice to set more than 4 or 5stations on curve from any one point . MR. SHUNK gives the l imiting angle to be deflected from tangent as and says 15° shouldrarely be exceeded. (Field Engineer, p .
1 00 . T he T ransit N otes m ay be conveniently kept in the formbelow , which shows the notes for the last example.When possible the tangents should be run to intersec tion,
the
angle 1 measured, and the tangent distance calculated. T hen
62 A FIELD -MANUAL FOR RAILROAD EN G IN EERS
Station. Rem arks.
Deflecti
on
ang
l
e
Index
readi
ng
Cal
cul
at
ed
Co
urse
Magnet
i
c
Course
l l
.
l l I l
+40 0 P .T . 0° 48 ’
33° 48:32° 36’ N E N E
21° 0’
19° 0’
17° 0 ’
15° 0’
5° 30 ’
2° 0’
1 ° 30'
4° C .L . ; P .I . set .
+25 0° 0 ’0° 0’ 0° 0’ 1 : 32° T :
ft .
N 60 12 E N GO° 10 E
m easure along tangents and set B C. and R T . from the P . I .
When the curve i s run ih , the posi tion of the R T . thus foundshould agree with the one set from the B I . If the error isgreater than the c i rcumstances of the case permit, the curvemust be rerun and tangents rem easure‘
d.
1 0 1 . Ano ther F orm ofN o tes, and in som e respec ts a bet ter onethan the above
,i s given below . T he index -readings are com
'
puted as though the enti re curve were run from the R C. The
notes for the las t example would appear as below
S tat ion.
+40 (DR E. 0° 48' l6°
é8 ’ 32° 36’ N 27°30’ E
15° 0'
5° 3C’
2° 0’
1 ° 30’
4° curve left ;+25 0° 0’ 0° 0’
T : ft .
N 60° 10’ E
T he computat ions are all made before beginning the work , and
the notes have the advantage of perm i tting the trac ing of the
curve ei ther way from the instrum ent withou t addit ional comp u
LOCAT ION .
tations. Suppose the transi tman to have run the curve from the
B C. to sta. 85, to which po int he removes the instrument . He
there sets the vernier at 0°— the angle on l imb when telescope
was in tangent at the P . 0 .— then sighting theR C . he reverses
the telescope and deflec ts to 9° which wil l fix sta. 86. Had
the tangent at 85been desired, a reading of 7°
30’— the angle that
located that point— would have put the telescope in the plane desi red. A reading of 11
°
30 fixes 87 , and so on to the R T .
Rem oving to thei
P . T . , the plates are c lamped at 7°
and a
backsight to sta. 85 taken ; then deflec ting to 16°
the telescope is in tangent at the P . T Had i t been desirable to set 84from 85, a reading of 5
°
30’ would fix that po int ; others m ay
be found in the same manner .~Any convenient fo rm of notes , which are intel l igible to another
engineer who m ay have to retrace the curve, m ay be used, bu t iti s desirable that some general fo rm should be employed. Eitherof the p receding forms seems to meet ordinary requirements.
0 . Obstacles.
1 0 2 . T o P ass an Ob stac le on a C urv e.
FIRST . S upp ose the obstacle to be one obstructing vision a t one
sta tion only .
1 11 Fig. 25 suppose transi t set at A , and B and 0 located fromthat point , bu t t he nex t ful l stat ion
,E
,to be invisible from A .
FIG . 25.
Set a plus stat ion at E , as near the obstruction as m ay be conven
ient,then set F 100 feet from E . Nex t make FG 100 CE,
and locate G with the corresponding deflection-angle. Otherstakes m ay be set beyond G, or the transi t m ay be removed tothat. point and the curve beyond traced.
SECOND . Suppose the line of sight obscured for m ore than one
sta tion,as in Fig. 26.
64 A F I ELD -MANUAL FOR RAILROAD ENGIN EERS.
If transi t i s at A , deflect an angle HAB that wil l clear all oh
struc tions, and at the same ti m e cause B to fal l at a ful l stat ion.
T hen by T able IV , T able IX , or by formula ( 16) calculate the
long chordAB measureAB and move t ransi t to B then deflect
FIG . 26.
the angleABC: BAH when the telescope wil l be in tangent .T he curve m ay now be run both ways from B.
If it happen that some stations, as E and E in the figure, are
st i l l inv isible, they m ay be located by offsets from chord or tan
gent .EXAMPLE .
- Let the curve be 8. 3° curve to right angle HAB
: 7°
the defiec tion-angle for 5 stat ions. By T able IV the
long chord is feet, which can now be measured and a hub
set at B ; then making angle OBA 7°
the telescope will bein tangent and the curve can be traced ei ther way .
T o L oc ate a Curv e when the P . C. is Inac cessib le.
In Fig. 27 let theR C. at B b e in
accessible i t i s desi red to reach a
point B on accessible ground.
FIRST METHOD . Assume a
point B on the curve such that al ine AH from an accessible pointA , on tangent , will clear the oh
siacle ; for convenience H shouldbe at a ful l station. T he are EH
and central angle, which equalsHOF, are then known. CalculateBC : T by ( 14 ) or (14a) ; thensince AB i s known, A 0 , AB+BC,
is known.
Now in triangleACH , from trigonom etry ,
tan g(h a) AC OH
tan g(h a ) A 0+ UH
66 A FI ELD-MANUAL FOR RAILROAD ENGIN EERS.
SECOND METHon.
— If F , any assumed point in tangent , i svisible from A ,
AF m ay be measured by some indirect method;then AR AB T . T he tangent for 8. 1° curve having same
intersect ion angle, K FG, is T , T X D ; find this val ue of T . in
T able IX“and take out the corresponding value of I . Wi th
transi t at F deflec t the angle K EG, m easure FG FB T and
set hub at G. The stat ion number of G is found by dividing thecentral angle, K EG, by the degreeofcurveD . Move to G and
t race the curve.
EXAMPLE .
— Let AR m easure ft . from sta. 139 of the lastexam ple. T henAB 9 25ft and BE : 225 ft .
x 4 1062 ft . , which by T able IX is the val ue of T 1 for
I : Set transi t at F , deflec t and measure FG ft .
L chains ;
hence G wil l fal l a t sta. Move to Gand run the curve both ways .T HIRD METHoo .
— In Fig. 28 let the inaccessible P . C'. be at B,
and let i t be required to reach E from a po int C on the curveprolonged backwards from B .
E At a given point A on tangent cal
culate the tangent offset by (36) orthe m ethods of 9 5,
then set th is off atright angles to AB ; set the transi t at
. a i 0 and turn off A OL 90° COB,
when the telescope will be in tangent
4b
at G. COB m ay be found from T ableY
IX by multiplying AG by the degreeof curve and taking half the intersec
FIG . 28 . t ion-angle corresponding to the m id
ordinate that equals this produc t . N ow deflec t and measureE CL ,
then by (16) or (16a) calculate CE , which measure. Moveto E and deflec t LEG EGL and the telescope wil l be in
tangent . The central angle BOE 2LEG BOG,from which
the arc BE and number of sta. E m ay be found.
EXAMPLE .— T ake the same example as in the last two cases .
.4 i s at sta . 139 , B at 141 25; hence AB stat ions .By z AO gX X 4 ft .
139ofthelast
llovetoG
Or by T able IX the angle co rresponding to the long chord(2 X x 4 z 1800 ft . i s 18 °
for which the m id-ordinate is4
ft . ,which equals A C and agrees closely enough wi th the value
for 2 above.
Make angle BAG: and measu re A 0 ft . Moveto C and sight to A , then make angle ACL 90
°
5 (9°
80°
Suppose an angle LGE 16°
1'
to clear the obstac le.
By formulaCE 2R sin ( 16
°
2 X X ft .
ft . For our 4° curve the m id-ordinate wil l be
Measure along GE ft . and set a hub ; move to E and run
the curve.
CE might have been found by means of T able IX ,for the long
chord of a 1°
curve having I °LGE 32°
2’ i s ft . ;
divide this by 4 and there resu lts GE ft .
1 0 4 . T o P ass t o T angent when the P . T . is Inac c essible.
T his i s just the reverse of the preceding problem , and m ay be
accom pl ished by reversing the processes desc r ibed above.When the P . T . , however , fal ls in or beyond a ri ver or lakeobstruct ing the ordinary methods of indi rec t measurement
, the
case meri ts a spec ial so lut ion.
FIRST MET Hon .
— Iu Fig. 29 let the transi t be at A , and B the
P. T . From the known stat ionnumbers of A and B the length of
curve and angle I m ay be foundthen, by AG: R tan %I , or
by ( 14a ) , AO
Move to C and deflec t the angleI ; set a stake F , and one at someother accessible point E ; measureangle ECc . Move to F and
measure the angle EEG and the
side ER; then in tr iangle ECF
angle e 180°
(c+f) ; by trigonom etry
FIG . 20.
sin 6
sin 0CF EF.
68 A FIELD -MANUAL FOR RAILROAD EN GIN EERS.
Since BG AG, there resul ts BF CF AG; and as the stat ion number at B is known, that at F becomes known, and the l inem ay be cont inued.
IfB is not the P . T ., measure back the distance FB, set transi t
at B, and continue the curve.
EXAMPLE .—Let the P . T . of a 2
°
CI . fal l at sta. 205 50— an
inaccessible point ; suppose A at sta. 200, angle 0 fEF 310 ft .
Here I x 2 1 1°
and e 60°
By (14a ) , T ftf
From applying logar ithms,log CF
Whence GE : ft . T hen BF : ft
therefore the num ber of F will be 206SECOND MET Hon.
- In Fig. 30, with the transi t at any po int Aon the curve, assume a long chord ABand calculate the angle GAB; deflec tthis angle from the tangent AG, and set
a point E beyond obstruc tion; set alsoa stake at G in tangent .Move to E and measure AEG and
'
s ideEC. Compute AE from the triangle AEG. If this is greater or lessthan the length of the long chord AB
,take thei r difference BE and set a hub
at B. W ith the t ransi t at B trace out
the curve.
EXAMPLE .— l en A at sta. 210 of a
3°
C . L .
,angle a z l 2
°
,6: E0
F‘G 30 18 1 ft . T hen 0 and by sol vingthe triangle A EC, AE : ft . By T able IX the long chord of
a 1° curve for I : 24 ° i s ft thereforeAB
3
ft . N ow will EB : ft . ,which is the dis
tance along EA that transit must be moved back from E .
LOCAT ION .
1 0 5. G iv en the P erpendic ular p from a P oint to a T angent ,
to Find the 'P oint on T angent at which to Begin a C urve of
G iven Radius which will Pass through the G iven P oint .FIRST 8 0 1.UT 10 N .
— In Fig. 31 let P be the point, BP the perpendicular. We have to find
From P draw PC paral lel toAB then in triangle OPO’
It 2 7° (R
From whicha: 4/2Rp (43)
“L" ‘
u,SECOND SonUT 10 n.
— Conside1' 2
p AO as the m id-ordinate for 5a long chord 273:then p x D Ethe m id-ordinate for a 1 ° curve V0
for a central angle equal 2a .
FIG ~ 31.
T he corresponding long chord m ay be taken from T ableT hen
- (43a)
EX_AMPLE .— Gi ven p z:30 ft D 4
°
(R to find 73.
By a: V85,962 900 feet .By the second method
30 X 4 120,
the m id-ordinate for a 1° curve corresponding to an angle of
23°
for which the long chord is Now,by (43a )
2: feet .1 06. In F ig. 3 1 , G iv en a:and p to F ind the Radius of a
C urv e T angent to AR at A and P assing through P .
2 -z
From (43)a: p
70 A F I ELD -MANUAL ron RAILROAD EN GINEERS.
1 0 7 . G iv en the L ocation of a P o int P referred t o the R I .
to F ind th e Radius of a Curv e through P which will Unitethe G iv en T angents .
FIG . 32.
In Fig. 32 suppo se BC Z, BP m known
,and angle a cal
culated or P C and a m ay be measured on the field.
From triangle 0 1 1 0 ,
b 90°— (a -l- g-I ) , and 0 0 : R sec é I .
N ow from triangle F CO,
COs in g ROsin b.
Insert ing val ues ofP 0 and 0 0 ,
sin y :
an equation from which the unknown R has disappeared. N ex t,
from the same triangle,since 2: 180
°
(b -l
P C .
When I i t can easily b e shown thatR Z + 7n t
'
2lm .
LOC ATION .
1 0 8 . T o Loc ate a T angent to a Curv e from an OutsideP o int .FIRST Mensou — In fi g. 33 1et P be the point and ABB the
curve. Run a trial - l ine PA cuttincr the cu rve in A and B.
Measure PA and AB :or m easure PA and angle 0 between the
cho rd AB and tangent A L . T hen
14 13 2 221 4 0 : 212 5111 0 ,
OC = R co s a .
By geomet ry ,PE t
’PA x PB, PE being the required tan
gent. From the figure,
tan n
tau m
At P deflec t the angle I m n from PA and run the tangent.SECOND hIE T HOD s
— In T able IX find the long chord for a
cent m l angle then
and CO = R — CH
We m ay now proceed as before.
72 A F IELD-MANUAL FOR RAILROAD'
EN GIN EERS.
1 0 9 . T o Run a T angent t o Two L oc at ed C urv es ofC ontraryP lexure .
FIRST CASE .
— In Fig . 34 let EX and LE be the curves,'
and
K L p measured on the ground.
FIG . 34 .
Let FE t be the required tangent .Draw parallel and OQH perpendicular to FE from thetriangle O.H0 2 since FII R.
(Ex' i‘ R2 + 1 0
2 (Ex’ i‘ R2 )
2 t?
whence
R1 + R2
Also , cos aB i +R2 +p
The arcs EX and LE m ay be found from the angle a and theknown curvatures, after which the points F and E m ay be set.
If t i s given and 1) requ ired,it m ay easily be found from
SECOND CASE . p not known.
Set the transi t at a point A on one curve and note the bearingof the tangent to the curve at that point (see Fig. the bearingof the radius O2A difi ers from this by Run a line ABC of
one or more courses to intersect the o ther curve at 0 . Note thebearings and lengths of these courses and the bearing in tangentat G,
from which cal culate the bearing of R. and R2 beingknown , the lat i tudes and dep artures are nex t calculated. Let 0 22V
(49) Cllit 't ‘
74 A FIELD -MANUAL FOR RAILROAD EN GIN EERS.
whence
R. R+p 2EF _ 100R
_ 100R +
R). (50)
Had EFG been the located curve, with radius R,we should
have had
AB = 100 o o o o o o o
1 1 1 . T o C hange the R C. ofa L ocated C urv e so that P . T .
will F all in a G iven T angent P arallel t o T erm inal T angent of
L oc at ed C urve .
A C G P Let AR,Fig. 36, be the 16
cated curve ; FE , the tangentin Which the P . T . m ust fal l .Let the distance between tan
gents be HE p .
Draw BE and 0 0’ paral lel to
AR ; evidently AO 0 0'
:BE ,
F101. 36.
0’ being the new position of
center.In triangleBEH,
LOCATION .
1 1 2 . T o F ind the C hange in Radius and P osition of P .G. if
P . T . is Required t o fall on the sam e Radial L ine but on a
T angent distant p from ,and parallel to , T erm inal T angent to
L oc ated C urv e.
In Fig. 37 let AB be the located and GE the required curve.
Draw the paral lel chords AB and
CE . Draw OH andBF perpendicularA C K
toAB . T heangles FBE:GAH=%I .
From the figure
CH : AC Sin 41 ,
' BF z BE cos eI z p cos alI.
Equating,. AG sin 4I : 1) cos {?I ,
whence Em . 37.
A0 = p cot §-I .
In the triangle GPO) , OiP AO, OP R 5 R1 and
(R — R1 ) tan I = AG z p cot g-I ,‘
or R
T hereforeR. = R — A 0 cot I z R —
p cot g—I m ot 1 . (54)
From trigonometry,sin I
1 — cos land cot Icot {51
Inserting these val ues in (54) gi vesR — R
sin I cos I cos IR
‘
cos‘
Ip '
l — cos l'
sin 1_
l — cos I.
iyet s I
'
From trigonometry, ex see I
ex see I'
76 A FIELD -MANUAL '
FOR RAILROA D EN GIN EERS.
EXAMPLE .
- 4A curve strikes 25 ft . inside 11 tangent inwhich the P . T . must '
fall . Find the necessary changein radiusand
‘
position'
ofP C. when IBy (53) the change in P . 0 . i s
AO 25x ft .
By R, ft .
By Tab le I we find this to be the radius of 5 2° 47' 41" curve.
1 13 . G iven a L ocated Curv e uniting Two T angents t o
F ind the C hange in P osition ofE U. or in Radius for a G iven
C hange in the Intersec tion-angle.
FIRST CASE — Radz‘
us unchanged.
InFig. 38 let BOE I be the origi
nal intersect ion angle, FCE I the
new angle. F rom the figure,
AG r - AU G0 ,
AG R (tan §I tan (55)
BY T ABLE IX .
— From the table, forangle I ,
E101. 38 .For I
'
T —
7;
T hen AG T T’
SECOND CASE .— P .C. unchanged.
Here the tangent T for the two cu rves IS the same, and
therefore
LOCAT ION .
BY T ABLE IX ,
T 1 4 Io
T 1
D D .
whence D ,M ! £ 4 1
T .4 1° T
1 1 4 . T o F ind the C hange inRand P . 0 . for a G iven C hange
in I , the T rem aining unchanged
FIG . 39.
In Fig. 39, from the tr iangles OBG and 0 1BH
0 G R cos I
and O.H z R1 cos L .
NOW GA HF ; henceR,
— ~R. R R cos I .
VVhence
1 cos I vers I .
R' R1 cos
_
I , TRvers I ,
’
'
Also , FA HG EH BG.
Insert ing values of.EH andBG, there resultsFA R sin I . (58)
1 15. G iven a L ocated C urve to F ind the C hange in R for
a G iven C hange'
in T ,I rem aining unchanged.
In Fig. 40 , from the triangles OA O and sinceEA 2
" EU AC ,
78 A F IELD - MAN UAL FORR
RA ILROAD ENG IN EERS.
R1 tan QI - R tan —gs A T'
T .
Whence R. R+ (T’ T) cot £1 .
FIG. 40.
BY T ABLE IX .— EA being known, 7
”T EA . T hen, by
T , 54 1°
1”
If the change in vertex of curve i s wanted,there results, from
E : CG T tan i], E'
CH : T'
tan il .
T herefore 0 151 E'
E : (T'
T ) tan (60)
GE can be found from T able IX after finding I) , as above.
IfR. is given and EA wanted, (59) y ieldsEA : T
’
T = (R,
1 16. T o F ind the Radius of a Curv e hav ing the Sam e P .U.
as a G iven C urve,but ending
'
in
a P arallel T angent .
In Fig. 41 let the perpendiculardi stance between tangents be p , and
AB be the located curve ; A O, R1
i s required.
FIRST MET HOD .— Draw OH atr igh t angles to 0 1E ; then
0 1H + HG + GE,
FIG . 41. R1 (E 1 R) COS I + R 1)
From which R1 12+1 —
p
cos I ves ’
LOCAT ION .
SECOND METHOD .
— A , B , and E lie on the same straight l ine,
since I is the same for both curves. In triangle BGE angleEBG 11 , and
pcosec 1 I .
em %Ip 3
From T able IX , ABL ' a t 4 1
AE’
AR BB i s the long cho rd for curve of degreetherefore
If desired R m ay be found by or T able I.
T HIRD METHOD .
-B l'
aw FL paral lel to OIE ; then
p cosec I .
Fro m T able IX A0
AF : AO CF , the tangent distance for second curve ; hence
A _fl i l
REMARK .
— Ii transi t. i s set up at B, i t wi l l be wel l to set E
by m easurement from B, to serve as a check when the curve isrun in from A .
80 A FI ELD -MAN UAL FOR RA I LROA D ENGIN EERS.
ART ICLE 9 . COMPOUND CURVES.
A. Location Problem s .
1 1 7 . G iven Two Unequal T angents, their Intersec tion-angle,and One Radius, t o F ind the O ther Radius of a C omp oundQurv e uni ting T angents .In Fig. 42 , AH : T 1 and BIZ : T , are the known tangents,
AO, R. the known radi us . B 0 2 R2 and the angles I , and I 2must be found before curve can be located.
FIG . 42.
Extend first branch to F , so that tangent FL is paral lel to EH.
Draw EX and BG perpendicular to FL draw FB and extendto E ; i t wil l pass th rough the because the cent ral anglesE0 ,F and EG21? are equal . T hen
8 2 A FIELD -MAN UAL FOR RA ILROAD ENGIN EERS.
By T able I th is i s seen to be the radius ofa 3°
7 -5'
curve.
T he length'
offirst branch i s feet , and of the secondfeet ; hence the P .C.C . fal l s at 112 while the P . T . i s a tsta. 120
1 1 8 . G iv en the L ong C hord from P C. t o P . T . of a C om
p ound C urv e, the Angles it m akes w ith the T angents and
OneRadius, t o F ind the O ther Radius and the C entral Angles.
In Fig. 42 AR i s known, as also the angles HAB a and
HBA 6. T wo angles and one side of the t riangle HAB are
known, and the sides HA T 1 and BB T 2 m ay be found,
after which the solu tion i s the same as in the last problem .
A sol ution m ay be reached in a difi erent manner. I z a b,
HAF : 4I : g(a b) , and BAF : 1}(a b) a z 40) a ) ,AF r: 2R, sin %L In tr iangle BAF two sideS '
and the inc l udedangle are now known, so BF and angle EFA m ay be found;GFB z. 41 2 4I BFA .
T hen EF 2R, sin 4L
and EB 2 EF BF becomes known.
T hen EB 2R2 sin 41 2 2R, sin 4L BF ,
whence R2 R1
Evidently I , I I 2
1 1 9 . G iv en the Radii and C entral Angles of a C om pound
C urv e to F ind the T ahgent L engths, the L ong C hord from
B C. t o P . I .,and the Angles it m ak es with T angents.
In Fig. 43 draw AE and BE from the
R C. . and P . T . to the thencalculate AE and BE by ( 16) or byT able IX . In triangle AEB angleAER 180 g(I , I 2 ) . T wo sidesand the inc l uded angle being known,
the triangle AER m ay be so lved forAR and the angles ABE and BAE ;then
LOCAT ION .
AB i s known, the sides AF : T 1 and RF : T 3 m ay be com
pu tca.
12 0 . G iv en the L ong C hord from B C; t o P . T . of a C om
pound C urv e and the Angles it m ak es with T angents t o
F ind the Radii When the C om m on T angent is P arallel to L ongC hord.
In Fig. 43 let GE be paral lel to AR, and GAB a
known. T hen
BAE RAG GEA 4a,
ABE EBB HEB 4b.
AER 180°
%(a b) .
In triangle AER, remembering thatsin [180 Ma. b)] sin 1}(a 4; 6)
Since AG1 1? a and E0 23 b, the radu R1 and R
found from formula or ( 16a) .
1AE AR sin 46By R:
sin .1c 2 sin %a sin % (a b)
.
R:133 5
”AR sin %a
sin 3126 2 sin %b sin % (a. b)
’
EXAMPLE .
-Required R. and R2 or D , andD , when AR900 feet , a bBY
By
From T able I, D . 2°
22'
50"
and D 22: 3
°
42’
8 4 A FIELD -MAN UAL FOR RAILROAD EN GIN EERS.
B. Obs tac/es.
1 2 1 . T o L oc ate a P oint on ..ne Second Branch of a C oni
pound C urv e when the P .C. C. is Inac c essib le.
Ordinari ly the second branch is located by setting transi t at .
the
P . C. C . and running the curve from that point . An obstac le on
ei ther curve m ay then be passed by the methods given for simplecurves.
FIG . 44.
are readi ly found thenEL R2 vers b R, vers a ,whence
vers b
AB = R1 8 in a +R2 sin IL (68 )
Deflcct FAB a from tangent at A measure out AR set thetrans i t at B and locate the second branch .
BY T ABLE IX .
— T ake the m id-ordinate in table for an intersect ion-angle 2a then
M1 4 2a.D
T hen EL x D , i s the m id-ordinate.
for a 1° curve having
I 20,from which 1) becomes known. From the table now find
AL and I R, the half- chords for angles 2a and 2b, and proceed as
before.
SECOND METHOD .
— By m eans of tangents .
From Fig. 44 , AF : FE R. tan 33a .
When the P .C.C. IS 1naccessible,
locate thefirst branch from theP .C.
and the second branch from the
P . T . ,if this lat terpoint i s known.When this 'i s not the case proceed
by one of the following methods:FIRST . By m eans of a long
chord.
In Fig. 44 let E be the.
A some known point on first
branch , EF a tangent at E , and
AR parallel to FE. T he stationnumbers of A and E beingknown, the arc AE and angle a
86 A FIELD -MAN UAL FOR RAILROAD EN GIN EERS .
Ri— Rc
—p
Rx— RQ
T hen a divided by D , gives arc BA .
If desi red, BE m ay be found from the right triangle BE G,in
Wh ich the side B0 p and angle GEB 445 are knownA ,H
,and B lying in the same straight l ine ; then
COS a
P 1cosec a .
$ 111 450p 75
Or BA andHA m ay be found from T able IX ,after which
EXAMPLE .
— A 3° curve ends in a tangent at sta . 160 50
,
35ft . outside ofdesi red tangent . Find the po int of compounding with a 4
°
50’
curve.
From T able I , R for 3° curve equal s ft . , and for
4°
50' curve ft .
T hen,by cos a 1
From table of cosines angle a is found t o be 17 ° D ividingthis by 3 gi ves sta tions for the arc BA . Hence the P .
number is sta. 154 and the new P . T . i sat sta. 158
1 2 3 . G iv en 21 L oc ated C ompound C urv e ending in a
T angent P arallel t o, and a G iv en D is tanc e from,a T angent
in which the C urv e is required to end. T o F ind the N ac es
sary C hange in P .C’
. 0
FIRST CASE .— Term z
’
nal branch ham’
ng shower radius.
In Fig . 46 let ABC be the locatedcurve, ABE the one required ; angleB0 1 0 a known, and al so MN r:19 .
If angle EOM b can be found, the
8angle of retreat from B to E wil l equalb a .
C Draw O.
’K and OJL perpendicular
to ON , which is paral lel to 0 , 0 .I
K L M T hen OE Z (R — R1 ) COS b,
FIG . 46.
0 L (R R.) COS a .
LOCAT ION .
Now EM: R, K L R, MN , from which XL Ms .
Hence
(R — R1 ) cos a —p .
From whichcos b cos a
Divide b a by D , the curvature of first branch , and moveback that number of stat ions from B to the new P . a t E.
Join evident ly F0 and angle K 0 1
'
0 , 0 170 ;
2 90°
% (b a) , 0 0 .
’
K 90°
b. Hence
CFG K 0 ,
'
0 , [90 g(b a )] (90 b g (b a ) (72)
From triangle OGE,
29
sin 20) a)p cosec g(b a ) . (73)
Or, from triangleF0 1: 2(R R1 ) Sill %(b a) .
Bad AEF been the original curve, b would haye been knownand a requi red.
From cos a cos bCF and angle CFM are given by formulas (73) andEXAMPLE .
— A 2° curve compounds wi th a 4
° curve at sta.
82 30 ; a 20°
p z:40 feet . Find number ofnew P . 0 . 0 .
and distance between P . T .s.
40From cos b
T his y ields b 24°
and b a 3°
The change in P . is3
233 stations; the P .0 . 0 .
number is therefore sta . 80
88 A FIELD-MAN UAL FOR RAILROAD EN GIN EERS.
By CFG 20’ 20° z 22
2
By F0 40 X feet .
SECOND CASE .— l
F IG. 47.
0 ,'0M and 0 .0L ,
(R, R) cos b 0 .
’
N + (R, R) cos a .
Bub Og
'
N z KB p ; therefore(R,
— R) cos a .
Whence cos b cos a
a — bT hen
Dwill be length of curve from A to E.
Angle K FB NO1 0 1
'
0 0 1 0 1
,1170 1 0 .
Bnt 90°
4m b) and N 0 , 0 90 a .
K FB [90°
t (a b)] [90 a] g(a b) .
From triangle K FB,
sin( 76)
Or, from t riangle since 0 , 1
'
FB,
FB 2(R, R) sin g(a b) .
term ina l branch having longer radius.
Let CAB, Fig. 47 , be the locatedcurve wi th P .C.C. at A , and let
EX be the tangent in which thecurve is requi red to end.
T he di stance BK z p , the radii0A R, 0 1A R1 , and angleA 0 1B = a being known,
i t w il lbe suffi cient to find angle EO,
’
F
in order to get the angle of ad
vance, AOE a b. Draw 0L
and 0 ,N perpendicular to 0 ,
’
F
and 0 ,B . From the triangles
90 A FIELD -MAN UAL FOR RAILROAD EN GIN EERS.
From tn'
angles K FH and E CG ,
HK GE GE1 l p
tan °b _
FH“
Go+173
tau °a +FH
But FH : (R R1 ) sin a .
(R B l ) Sin (1
From triangles and
(R R2 ) sin b (R R1 ) sin a .
When R9 R (R Rx)
Had AEF been the first curve located, b and R, would beknown,
a and R, requi red.
From the figure, reasoning as before,
(R Ra) Si ll b,
R, : R - (R — Rg)
SECOND CASE .- Second branch having longer radius.
FIG . 49.
In Fig. 49 let. AB be the located cnrve, EF the curve requim d,
0A 2 R, 0 1A 2 R1 , OQE : R2 , FB 2 p .
LOCAT ION .
R, and angle b are wanted, angle a being known.
We can show,as in first case , that
1120, HBL ia,
0M : K F : LB : (R.— R) sin a ;
and hence
tanEX HL p
Or insert ing values,Pltan gb tan 4a
(R, R) sin a.’
Angle b now becomes known and AE in chains,which
i s the change in posit ion of P . 0 . 0 .
From triangles and
(Rg
— R)
'
Had thenew tangent fal len ou tside the old one, we should havehad
_ L _ _
(Ra— R) Sin b
’ Otan tan 4b+
— R) c o o1 25. Having a L ocated C om p ound Curve
,to F ind the
C hange in P . and Radius of Sec ond Branch in order to
C ause P . T . t o F all at a N ew P oint in T erm inal T angent .
FIRST CASE .
— Second branch lea ving shor ter radius.
92 A FI ELD -MAN UAL FOR RAILROA D EN G IN EERS.
In Fig. 50 let NAB be the located curve, and C the po int whereP . T . i s required to fal l . Let B 0 k , 0A R, 0 ,B R1 , and
angle 0 ,OH a be known; angle b and R2 are required.
FIG . 50.
Extend fitst branch to F ,m aking OF parallel to 0 ,B. A
, B,
and F lie on a straight l ine, for angles A 0 |B'
and AGE are equal ;l ikewise E , 0 ,
and F lie on the sa m e st raight l ine.
From triangles GBF and 0 CF,
cot gb .
But GF EM (R R1 )(I cos a) (R R, vers a .
1 1
(R — R,) vers a'
From triangles and OOQL , since OIP k,
(R R,) sin b (R R1 ) sin a k.
Whence
k ‘ (R — E j ) sin a
sin bT hen b a di v ided by D gi ves arc AE . Wi th radius R2 locate
the curve E 0 fro m 0 or E .
94 A F IELD - MAN UAL FOR RAILROAD EN GIN EERS.
Had NEG been located andNAB required, the equat ions wouldhave been
kcot 4a z cot .
‘gb
—t—(Ra R) vers b
’
Rg— R SIn b k
In either of these two cases ifk i s unknown and the new radiusgiven or assumed, the desi red angle and the val ue of It m ay be
found from the foregoing equations . Or, knowing the new angle,
the new radius and value of 16 m ay be found from the sam e
equat ions .1 26. T o Replac e a C urv e of G iv en Radius
,which unites
Two T angents with K nown Int ersec t ion-angle, b y a T hree
c entered C ompound C urv e.
In Fig. 52 let 0A R be the radi us of located curve
FIG . 52.
0 2 0 2. Og
'
A R, the radiu s of terminal portions of the threecentered curve,
and the other notat ion as shown in the figure.
D raw and draw FOH perpendicular thereto . From triangles 0 20 13 and 0 20 11
0 211 : (R2— R,) sin 5
1 1 ] (Ra R) sin é I . (a )
SupposeR, and R, to be assum ed then equat ion (a) y ields1
SID f ]: o 0 O
LOCAT ION .
T hen x 4a L ) .
Suppose AOQ’E, 0 0 2 0 ,and R2 to have been assum ed. From
( 95) find I ) then,fro m equation (a ) ,
sin 712 1
R. R2 (R. B) sin $1 1.
EXAMPLE .
— G i ven a 4°
curve, I : and the terminalb ranches composed of a 2
°
curve for two stat ions, to find R. and
D , for the central portion.
Here I . 38°
From T able I, R:z: 2865ft .,R ft .
Whence R, R ft .
Log 1432.3
sin 19°
Sin 15°
0'
log
T herefore R1 2865 1063 3 ft . , and, by T able I ,
1 5°
nearly enough .
1 2 7 . T o Sub stit u te a C urv e of G iv en Radius for a T angentuniting Two Curv es.
In Fig. 53 let the tangent B0 : t, 0B = R, R1 ,
and
0 1A R; be known.
Angles a ,b,and a must be found in o rder to substi tute curve
AE for the system ABOE .
Draw OF paral lel to EU, then R,
R, and , from triangle
t c osec ( l 1/ (R,R)
2 t’ . (98 )
96 A FIELD -MAN UAL FOR RAILROAD zEN GIN EERS .
Now in triangle three sides are known and the anglesa and 6 m ay be compu ted. T hus if 3 i s the half-sum of the sides ,
cos 50 2
FIG . 53 .
Angle 6 m ay be found in l ike m anner , then b 180°
(6 d),and a z c b.
Po ints A and E m ay now be located and the curve t raced.
EXAMPLE .—A 3
°
and a 5°
curve are uni ted by a tangent 500feet long. Replace by a 2
°
curve.
Here RI
R z 1910 1 146 764 feet .
By tan d tan 33°
12'
By 0 0 1 feet .
In triangle 0 0 ; and 0 0 2
feet . So lv ing for e and c,
e 2 133°
o 23°
T hen b 13°
a 2 9°
56'
ART ICL E 10. TRACK PROBLEMS.
1 2 8 . Rev ersed C urv es should never be employed on mainl ines because of the shock due to sudden reversal of curvatureand superelevation of outside rai l . A short tangent should beinterposed between the two cu rves, which m ay ordinari ly bedone by changing the end-points of t he curve, or sl ightly al teringthe radius . If, however , transi tion curves are em p loyed to ease
98 A FIELD -MAN UAL FOR RAILROAD ENGIN EERS .
EXAMPLE .
- A 1°
curve i s cut by a tangent that makes an angleof64
°
32' wi th tangent to curve. Uni te by means ofa 4° curve.
By cos b cos 76°
07'
and therefore b a
1 1° mak ing AF, of figure, stat lons.
SECOND CASE .
— Joim‘
ng curve tangent° internally to loca ted
curve but on app osite side of cutting line from center of loca ted
curve.
In Fig. 54 let arc ME ,wi th center 0 2 and radius R2 , be thejoining curve. From the figure
cos d :
T hen arc AE a d divided by D ,and o 180
°
d.
Had the po int E been given and R2 requi red, it would havebeen,
from (101)R(cos d cos a )
1 cos d
EXAMPLE .
-T ake the sam e example as in first case. Here,
By cos d cos 24°
T hen 64°
32’
24°
56’
39°
equi valent to stations around curve from A to E .
T H IRD CASE .— Joining curve tangent externa l
ly to loca ted curve,
with center on sam e side of cutting Zine.
FIG . 55.
In Fig. 55let arc B0 ,with center 0 1 and radius R, b e the join
ing curve. Draw 0 .E parallel to CF , and 0 , 0 and OF perpen
dicular thereto .
LOCAT ION . 9
From the figure,
— R1 ;
T hen d 180 b, and AOB b a . T he curve m ay now betraced on the ground.
IfAO i s wanted,we have AO (R R, ) sin b R sin a .
If the point B is fixed and R1 requi red, there resul ts ,’fromR (cos a cos b)
1 cos b
EXAMPLE .— T ake the example given for the first and second
cases .
By
5730 X
5730COS 8 1 44
b a z 8 1°
44’
64°
32’
17°
to
stat ions on located curve from A to B. Angle d _r: 180
°
8 1°
44'
z 98-0
equ ivalent to stat ions from B to 0 on the
4°
curve .
FOURTH CASE .
— Joz’
nz‘
ng curve tangent externally to located curve,
with center on opposite side of cutting Zine.
Let 0 2 , Fig 55, be center of jo ining curve, R2 its radius .From the figure,
R cos a + R2
R + R2
o 0 0 0 0 0 0 0 O 0
It'
M is fixed andR2 required, (105) yieldsR(cos c cos a ) E(cos c cos a)
1 cos c versin 6
EXAMPLE — T ake same exam ple as in preceding cases.
By cos c cos 57°
100 A FIELD -MAN UAL FOR RAI LROAD EN GINEERS.
cal l ing for a distance of stat ions;from A to M around
1° curve. From M to H on 4
° curve is stations.
1 30 . T o L ocate a Y
A Y i s made up of a system of tracks so arranged as to admitof turning an ent i re train. T hree of the most used arrangementsare gi ven below .
Fm s'
r CASE .
— 0ne branch of Y a straight line.
T his is only the spec ial case of the last problem in which thecu tt ing line becomes tangent to both curves . In Fig. 56, if any
FIG. 56.
one of the points A , B, or 0 i s given, the others m ay be locatedby finding the angles 6 and b. Draw 0 ,E paral lel to CA ; thenin triangle 0 0 1E
—.R1 .
R — R.
—
R+ RJ
T his fol lows at once from ( 103) by making angle a 0 . T henangle 6 180 .
b. If AB were a located curve and the pointB gi ven», formula ( 107) would furni sh us a value for R) .
Another solution is to produce the tangent at B to cut A0 at F ;
then AF F0 BF. Join F with 0 and 0 , it can easily beseen that angle OFO, and, by geometry
BF Z VR X Rl o o o o o 0
T herefore tan %b _
tan %0 0 0 o o o o o
thefts! 1'
102 A FIELD -MANUAL FOR RAILROAD ENGINEERS.
is the same as for second case. T hen b i s the central angle forcurve AB
,a'
1 80 a, the central angle for AO, and c
180 c , the central angle for curve B0 .
FIG . 58 .
EXAMPLE .
— If A i s at sta. 820 on the 1°
curve AB, AO an
8°
curve, connec t wi th a 6° curve 0B. Here we have5730 7 17 z 5013 , 0 . 5730 955 4775,
955 717 1672.
Solv ing this t riangle, we get 6 88° b 19
°
anda 72°
T he number of B i s therefore 820
839 the length of 0B is stat ions, and
ofA 0 i s1078 stat ions .
1 3 1 . T o L oc ate a Rev ersed C urv e b etw een P arallel
T angents .
FIRST CASE .— Radt
'
z'
equa l. F’f .
r;
(a ) T he equal radi i R and distance p between tangents known.
In Fig. 59 draw OE paral lel to A 0 to meet 0 ,B produced.
From tr iangle GEO) ,
COS G
OE : OR sin a
LOCAT ION .
From triangle ABG,
p cosec 40. MOE ?p’
FIG . 59.
(b) AG and p known,R required.
Here AB VAG" p" k . Draw OH to the mid point of
AO. T i'iangles AOH and AEG are similar and AH 416.
T herefore
whence
EXAMPLE .— Connect two paral lel t racks , 30 ft . c . to c . by a 7
°
reversed curve. From Table I , R 8 19 feet , and, bycos a 1 cos 10
°
By OE 1638 x . 19052 feet.By AB feet.If p 30, OF or AB had been given, we
should have had, by (1 15)R
1208 19 feet.
104 A FI ELD -M’
ANUAL FOR RAILROAD ENGINEERS.
SECOND CASE .—Radiz
°
unequal.
(a) Suppose the radi i R 0A and R1 G,B (Fig. 59) to beknown. We must find central angle a and AB h. From thetriangle
COS a
T hen AB will be given by(b) Suppose AB: k
, p and R known, to find R, and angle a .
T riangle ABG y ields
0 ,LB i s similar to AGB. Hence
But A 0 2R sin 4a ,and LB 401: A 0) 40 1 . Insertingthis val ue ofLB and solving for R1 ,
From similar tr iangles,R k 0 1
from (118) and solving forInsert ing th e value of 0 1
R1 we get
EXAMPLE .
— AB p R 8 19 ft . , to find angle
a and R1 .
By sin 42 sin 5°
T herefore angle a 1 1°
By R1 8 19 z 68 1 ft . , an 8°
25' curve.
106 A F IELD -MANUAL FOR RAILROAD EN GIN EERS.
1 33 . T o Find the Radius of the Rev ersed C urv e AFE ,F ig.
61 , G iv en Angles I and I’
, and
EU 2 70.
From the figure,
R tan %I BF ,
R tan OF
Adding,
R(tan 41 tan B0 : 70.
VVhence
ktan é I —k tan i l
"
EXAMPLE — GIVGH I 130 700 feet, to find R.
1 3 4 . T o L oc ate a Rev ersed C urveb etween Fix ed P oint s.
In Fig. 62 let AB k, and angles 1 and I be known. We
have to find R and the angles a and b.
LOCAT ION .
We now have a = I + w and
T o find Rwe have AE+ EF + FB : 1c,
R sin 1 + 2R sin fc + R sin I’z h.
Whence sin 1 + sin 1 ’
+ 2 sin cv”
Another exp ression for R can be found by drawing AN and BL
perpendicular to and EN paral lel thereto. T hen,since
4 BAN = 9 3
R sin a + R sin b = k cosw.
h cos w
EXAMPLE .-T ake the example of the last problem ,
k 700, I : I'
By
cos {3 g(0.98481 cos 15°
We now have a 25°
48’
and b 35°
700 xBy ( 128 ). R
m 523 3F0
1 35. T o C onnec t Two D ivergent T angents by a Reversed
'3‘urv e .
FIRST CASE .
—Adbancing towards theR I .
G iven the radu R and R) , the angle I and AO k, to find the
angles a and b (Fig.
ft an 8°
41'c urve.
L
FIG . 63.
Draw 0 0 paral lel to the tangent B0 to meet 0 ,B produced.
'l‘
hen EF B0 AF AE.
T herefore B0 R cos I k sin I .
108 A FIELD -MANUAL FOR RAILROAD EN GINEERS.
From triangle 0 0 ,G,
R1 + BG R1 + E COS I — k SIn I
R+ R. R + fl
T hen a MOJV'
b I, 0 ,M being paral lel to 0A .
SECOND CASE .—Recedz
'
ngf7'
0 m theR I .
In Fig. 63 we have B0 k , angle I , R, andR. given, to find
angles a and b.
P roduce 0A to meet 0 ,L drawn paral lel to CA . AL equals0 1M : G.H cos I .
’
Oxfl l — m l — k1 tan l .
AL : 0 1M = (R,— k, tan I ) cos I .
cos b ._
Hence
0 13 R —i— (R, k, tan I ) cos I = R+ R1 cos I k, sin I .
From triangleGE R+ RI COS I — k l sIn I
om R + fl
Evidently , b a I .
1 36. T o Change the RR. 0 . so that Sec ond Branch of
C urv e shall E nd in a T angent P arallel to T erm inal T angent
and D istant p therefrom .
In Fig. 64 let MAB be the located curve, EN We must
FIG . 64.
determine the angle COA , after which the desi red curvem ay be located.
Draw HO,
’
and LO. parallel to EF and NO.
HL 2 o |s .
CHAPTER IV
TRANSI TION -CURVES .
ART ICLE 1 1 .— THEORY OF THE T RANSIT ION -CUBVE .
1 3 8 . E levat ion ofOut er Rail on C urv es .— T o counteract the
efiec t of centrifugal force on curves the outer rail must beelevated above the inner one. It is shown in mechanics that thecentr ifugal force is
where W i s theweight , 0 the veloc i ty in feet per second,
an average val ue of the acceleration of gravi ty in feet per second
per second, and R the radius in feet .In F ig . 66let the vertical HL represent W, the horizontal K H
the centrifugal force, AB the plane of the rai l s , and CE e
the superelevation of outer rai lK H‘ From simi lar tr iangles ,
eF 2 W oW
AC
Equate th is value ofF to that given8
above and solve for e, gi ving
e
FIG . 66.32 . 16R
T he gauge AB should be greater on curves than on tangentsto al low for flange clearance and the efiec t of a r igid wheel -base.
AO feet i s abou t the righ t val ue for the horizontal distancebetween centers of rai l -heads for standard gauge. In formula(133) 0 is in feet per second , but the train veloc i ty is usually gi venin m i les per hour. Let V veloci ty in miles per hour, then the
110
TRAN SIT ION -CURVES.
22
15V. Insert m g theseveloc i ty in feet per second wil l be 0
val ues in ( 133) gives
><484 V°
I’
Z near]32 . 1ex 225R 3R
’ y'
T his elevation will be required from the P . 0 . to the P . T . , bu tobviously it cannot be introduced suddenly , so that for easyriding the rate of increase of e should be uniform . From ( 134) itis seen that e varies inversely with R, which requi res that whene 0, R infinity . Hence R m ust dec rease from infinity to
the radius of the circular curve, whi le e increases from to itsmaximum value.
1 3 9 . T he T rue T ransition-curv e should satisfy formulabu t so far no such curve has been found that w il l at the sam e
t ime admi t of the sam e case of locat ion as the si m ple c i rcularcurve. According to Rankine the first use of any other thanthe c i rcular curve was made by Gravatt about 1828 or 1829 ,
the curve employed being the curve of sines . Another methoddescribed b y Rankine i s attributed to Will iam Froude about1842 ; th is c urve was worked up in the Engineering N ews byA . M. Wel l ington in 1890. Other approxi m ations are the Ra il
road Sp ira l, developed byW. H . Searles in 1882, and the cubicparabola, described by C . D . Jam eson and E . W. C rel l in in the
Railroad and Engineering Journa l, 1889 .
In 1880 El l i s Hol brook described in the Railroad Gazette the
true transi t ion- curvc app l icable to s m all angles and short lengthsof the cu rve. In 1893 C . L . C randal l publ i shed form ulae andtables appl icable to large central angles for both the ofi set and
deflec t ion methods .
1 4 0 . T he N ot ation here employed will be explained w ithreference to 'Fig. 67 . T he curve 0BB’
0’ is the c ircular curve
ofi set at 0 and 0'
from the tangents by the am ounts OH and U’
H’
.
AGB and B'
G’
A'
are the transi t ion-curves . A is theor point of transi tion-curve, 0 the R 0 , B the B
’
the
P . T 0 . 0 the P . T . , and A'
the T he co-ortliuates of Gare A l l : as
'
,HG y
’
; of 0 , 23’
and EU F ; of B , AM : 1171
and MB T he length of curve from P. T . 0 . to any point Pis l, and theWhole length from P . T . 0 . to P i s 11 .
1 12 A F I EL D -MANUAL FOR RAILROA D EN GIN EERS.
1 4 1 . E quation of T ransit ion o curv e .
— ~ Since the rate of
change ofe must be uniform , (134) m ay be wri tten .
e z c
FIG . 67.
in which 76 i s the rate of rise of outer rai l along curve, and p thevarying radi us of curvature. From the calculus p dqb dl,
whencedl
2160 o o o o o 0
Insert this . in ( 135) and solve for d¢.
3kdgb 7,
—
2l 2777l .
2m is dependent upon V and k, and i s constant for any one
curve.
Integrat ing( 138 )
the constant of integrat ion being zero, for l i s zero when (b iszero
1 14 A FI ELD-MANUAL FOR RA ILROAD EN GINEERS.
Substi tut ing m l 2 for (p and integrating
m°l‘ 772418
10+
216(B:
Replac ing m l 2 by 0? reduced to degrees45 (P (b
x32828
“L2328 x 106 33114 x 10
(142)
E varies w i th and m ay be taken from T ableXIV wi th 05°as argument .1 4 2 . T he T ransition-curve Angle I , i s the val ue assumes
at the FromI ) m ll
zo o 0 o o o 0
From (137) and ( 136)
At the P . p R and m ay be taken equal to 57fl so thatD ° ’
whence
2l tR 1 146051.
T his value of m in ( 143) gives
Reducing th is to ci rcular measure by wri ting I , z I .
z. D°
l.1 .
200.
1 4 3 . T he C oordinates of any point on the curve are given by( 140) and T he length of the t ransi tion- c m
'
ve being known
TRAN SIT ION -CURVES.
or assumed, y. and re. (the cob rdinates of the P . m ay be
found from these equations by the help of T able XIV ; thecoordinates of the P. 0 . (see Fig. 67) wil l be
E z y , R(l cos L ) z y , R vers L , (147)w
,
z 12 8 111 I ) . o o o o ( 148 )
1 4 4 . D eflec t ion—angles .
— W i th the t ransit at the P . T . O. (or
P . T . . in backing up ) the tangent of defiec tion-angles m ay be
found from the telation tan 6 D ividing ( 139) bym l
”
tan 6 —
30 09523771316 .000167m 5l l o ( 149)
From trigonometry the expansion of the angle in terms of itstangent is6 tan6 T
1,
- tan3 6+ % tan56— etc . (a)
In (149) wr ite m l2 z and subst i tute in (a)3
o o o 0
From (138 ) and<2 m l?
L m l le l l?
O 0 0 0 0 0
in whichz
l71 . From (b), 05z Ln", and this in (150) gives
1
6 .002823I 1 3n6 (0)
Both 6and I , are in circular measure ; to reduce to degreesmultiply by
120 . T hi s gi ves, neglect ing term s invo lving higherpowers 0 1 1 , than the third,
I lo
3n°
T he second term is qu ite smal l , and in most cases m ay be en
tirely neglec ted in practice.
1 16 A F IELD -MANUAL FOR RAILROAD EN GIN EERS.
Wi th the instrument at any intermediate po int ce"y the deflect ion-angle for any point m y, measured from init ial tangent, wil l betan 6 i ZH % (m l
‘2m l
”?m ll
l l
) +T 0 5(m316 m
3l"5)
maufl s)+ 145111
3141” (152)
inwhich powers ofm l‘2 higher than the third have been neglected.
Subst itu te the value of tan 6from (152) in (a) , write m l2 z
L 779, m l
"9gb” by (b) , and reduce circular m easure to
degrees, giv ing80 % (n’ n
'”nn
”) a smal l correction. (153)
For instrument at n 0 then (153) y ields(60
—n° correction,
( 154) is the same as as it should be.
For the transi t at the quarter-point of transition-curve—l"
49 11111311 053) elds
1,5“
+ ln 417) correction,
— Bi o o o 0 o o o o o oFor t ransit at m id-
point of transi tion-curve n” 4, and, from
% (n’2 1+ 4 477) correction,1 1
°
'
H’
é‘ Ai
— B} o o o o o o 0 0
1 18 A F IELD-MAN UAL FOR RAILROAD ENGINEERS.
T able XV gives the values ofA and B for the five posi tions ofinstrument for which equat ions ( 154 ) to inclusive
,were de.
duced. T he val ue ofA must bemul t ipl ied by but B is taken
di rect from the table in thousandths of a degree.
Ifdefiec tion-angles are wanted for o ther posit ions of the instrument
,or for o ther po ints on the curve, they m ay be compu ted
from equation1 4 5. T ab les.
—T hree tables are given for use with transit ioncurves .T able XIV was computed for usewi th formulas (140) and (142)
in determining 0 and E (1) being assumed and 0 and E com
puted.
T able XV gives A and B for compu ting the defiec tion-anglesby and ( 159) for 20 equidistantstations on the transi t ion-curve. For points not given in thetable A and B must be interpolated. L inear interpolat ion wil lsuffice in most cases, though when I 1 ° i s quite large second differences m ay be preferable for A . B is gi ven in the table in thousandths of a degree.
T able XVI was cal culated by assuming I, in lengths varyingby increments of 20 feet, then computing I 1 ° by y , by731 by F by and az
' by y , and (0 , will also begi ven more di rectly by (140) and (142) with the aid of T ableXIV .
The excess in length of transit ion- curve, measuredfrom P . T .0 .
to the point on offset at P . 0 . ,over (13’ i s tabulated as e; l
’
is
found by trial such that when inserted in ( 141 ) or (142) the sameval ue of cc
' will be obtained as in T his m ay be done by as11
2
trial s wi l l rarely be needed to find a sufii ciently close val ue of l’
then e l’
cc'
. y’ i s found by (139) after finding l
’
, or 05
m ay be found from (b) of 1 4 4 ,and used in (140) in connect ionwith T able XIV . Z 1 l
’ i s the length from 0 (Fig. 67) to the
th'
e difierence in length between this and the length of
circular curve from P . 0 . to P . is tabulated as e’ that is
, e’
( l, l’
) arc . T hen e e'
Z 1 (z’
c i rcular arc) .For values of Z, interm ediate between those gi ven in the tablel inear interpo lation wi ll suflice,
though second differences m aybe used for F and y , if preferred.
sum ing l’
a l ittle less than then com puting x‘. More than two
TRAN SIT ION -CURVES.
1 46. T o Unite the Two Branches of a C om pound C urv e by
a T ransition-curv e.
T he sam e ob ject ions hold to compound cu rves as to simplecurves uniting with a tangent ; i.e. , where there i s a suddenchange of cu rvature there should be a sudden change of superelevation of outer rai l , which of course is not al lowable. Insteadof compounding the curves , we m ay offset them at the P . 0 . 0 .
and unite them by means of a portionofa transition-cu rve tangentto each of the simple c urves .In Fig. 69 AB and CELM are the simple Curves that are to be
uni ted by the transi tion-curveANE . Extend the transition-curveH W P
FIG . 69.
to G, where i ts radius of curvature becomes infinite, and let 0 8
be i ts tangent . Cal l the length of transition-curve from G to Al, from G to E la , and from E to A lg. E and A are po intsof tangency of simple and transition curves . T hen 1; l , —
‘
ls
T he cobrdinates of A are GS £13, SA z y . and of V (WVperpendicular to GW WV F.; of E
, GP x1
EP ya of L (LH perpendicular to GB HL F3 .
Let B0 F2 .
T he radius of curvature of t ransi tion-curve i s inversely proportional to i ts length from G ; hence the curvature i s proport ional to the length of curve; therefore la l, D 3 D , whence
120 A F IELD -MANUAL FOR RAILROAD ENGIN EERS.
D , D , D 3
T hen Z2 l l 63 l ) 1D 1
Z 1 D 10 (161)
By ( 138 ) 0 1‘
I I
Equating the value of I s from this equat ion to that resultingfrom (146) gives
WV : F , and HL F , m ay be taken from T able XVI wi thl, and I, as arguments . T hen 0 1W: R, F , 0 33 : R3 +F1 .
Draw 0 , T paral lel to GS, then 0 1 T WH hence
0 3 77 : (R3 F 3) (E 1
"i“ F1 ),
Ot T 931,
TB'
.
Therefore(U1
, — m a'
(Rs F 3) (Rx’ i‘ F 1 )
(x,’
fes’
) cosec a 1/ 0 1 T2
T O,2
(164)
Fz s o o o o
The lengths ofAB and 0E are
Lo
_ ao
a° — I 3
The excess of transit ion-curve length over AB CE is
eq z lg
122 A F I ELD -MANUAL FOR RAILROAD EN G IN EERS.
When only a sho r t tangent intervenes between two cu rvesshorter transitiom curves must be taken, requi ring larger val uesof If
, so that overlapping m ay be prevented.
For i l lustrat ion suppose 11 5°
curve to be eased offwith a transition-curve, the highest t t'ain-speed being 45miles per hour and
h z —1
By the table the value of Z 1 will be 7 1 X 5z 355feet600
so that we should probably take a 360-ft . t ransit ion-curvc, re
quiring an offset of feet by T able XVI.
ART ICLE 12 .
— FIE LD -WORK .
1 4 8 . For the cases most frequently present ing themselves inpractice the forego ing formulas m ay be sim plified so as to admi tof the rapid location of points on the transi tion-curve with all theaccuracy needed on location, though i t i s best to use the exac tformulas and tables in setting t rack -centers on the finished roadbed. When the t rans it ion-curve angle i s qui te large i t w i l l bebetter t o use the accurate methods on locat ion al so ,
but for the
more com m on cases the following formulas will answer .
1 4 9 . Sim plified P orm ulas .
— In (139) and ( 140) neglec t , as
smal l , all the terms fol low ing the first , givingl :
In (141) and (142) retain only the first two terms53 l 1 l (171 )
in which the las t term i s smal l for short t ransitione curves and
m ay often be neglected,(1:being tahen equal to l.
T he values of m and I remain as before
2RZ, 1 1460l ,’
o o o
TRAN SIT ION -CURVES. 12
1 1°
1 1 4
24 72 0"
But R by (145) Subst i tute this for R and neglect all1
But 3y, ,by since qb I , and y y , when I l,
hence
L 3
Rw m ‘6 120
11andwrt ttng R z: as above,
21 ,
l l l l l-
lg
w w) 5 12o o o
1 1 2 z 7,
m'
z,‘
16 12
‘
2
‘
2(nearly) . ( 173,
By
m3
m
X413 F
3 3 8 8 2
In and (159) neglect the correet ion; then
124 A F IELD -MANUAL FOR RAILROAD ENGIN EERS.
?A4) 0
?A} ) o 0 o o o 0
2 —'A 1 0 o o 0 o 0 0
(5c°
)
1 50 . Ofi'
set s.
— Formula (170) shows that offsets from transit ioncurve to tangent vary as the cube of thedistance from the P . T . 0 .
and i t can be shown that offsets from the circular curve to transit ion-curve fol low the same law, reckoning from the P .
Formula (36) m ay be wri tten2 f(l
Q-D) ’ o o o o o o o
inwhich D i s thedegree of curve if offset i s from tangent, and thedifference ofdegrees ifoffset is between two curves having a com ~
m on point of tangency , I being reckoned from the tangent-point .From (136) and
dl 1'
dgb 2m l’
and the degree of transi tion-curve at any point is114607721 Cl. 0 o o 0
Formula ( 18 1) shows that the degree of curvature of transit ioncurve at any point is a funct ion of its length . If the D in (a) i sthe difference betweendegrees ofcircular and transi tion curves, i twil l equal D , D , , which i s al so a funct ion of the length ; soin (a) write D f(l) , giv ing
o
which shows that the offset between circular and transi t ioncurvesvaries as the cube of the distance from T he offset at theP . 0 . is known,
being half ofF , and m ay therefore be found for
126 A F I ELD -MANUAL FOR RAILROAD ENGIN EERS.
(see formula ( 173) and set a stake, marking i t P . T . 0 . From the
P . 0 . measureforward around the c ircular curveadistanceequal towhich approx imately equal s Set a stake marked
At the quarter-point offset from tangent an amount equal to 414 1?
for, by the ofi sets areproport ional to the cube of the dis
tance from so thatF (4l, )3 1
2 (41, )3 16
A t the three-quarter point offset the same amount from circularcurve. If the transi tion-curve is not over 400 feet long, these areall the po ints need it longer , other ofi sets are similarly found.
EXAMPLE .
-At sta . 412 an offset of feet was made from atangent to a 5° curve. Requ ired the data for a transi t ion-curve
to connec t tangent and circular curve.
By T able XVI i t is seen that a 340-ft . transi t ion-curv’
e is re.quired. From the table it i s seen '
that az'
ft . , I ,°
and excess of curve over tangent is . 02 ft . , which we neglec t assmall . Drive a stake ft . from offset hub and mark i t 412 ;measure back along tangent ft . to 4 10 and drive a
stake m arked P . T . 0 . Measure forward around circular curvechains 170 ft . , and set a stake marked P . at sta.
413
T he approximate ofi sets areAt m id-point
,sta. 4 12,
one-eighth points , stas.fig1£2 t
quarter-points, stas . figi t z 0 , 033X23z
41 13three-eighths po ints , stas .
Stakes at the one-eigh th and three-eigh ths points were not
needed,bu t were worked out for i l lustrat ion
154 . L ocation by D eflec t ions.
— The nu m ber of chord-lengthsbeing taken as an al iquo t part of20, the deflec tion angles for the
TRAN SIT ION -C URVES. 12
transi t at any one offive posi t ions m ay be taken from T ableXV bymul tiplying the tabular values ofA by I ,
° being found from
Table XV I or form ula If the nu m ber of chords is not anal iquot part of 20, or it‘ the transi t i s at some point other than one
of the five for which T able XV was calculated, then the deflect ion-angles must be computed by The curve i s then run
out in the usual way .
When I , is not more than 15or 20 degrees the curve m ay be
run from the P . T . 0 . or P . T . by neglec ting the correc tionB as
s m al l . Even when I , i s greater than 20°
the correct ion m ay be
neglected,provided half the transition-curve i s run from the
P . T . 0 . and the remainder with the transi t at the m id-po int , thetelescope being first placed paral lel to original tangent .EXAMPLE .
— T akc the example of the last sec tion:I, 340 ft . ,
340 X 5F ft D : By formula I ,
200the
sam e as given by T able XVI . T henI
S), 2 D iv ide 11 into
5parts of68 ft . each , which wil l b e the chord- length to b e used.
Fro m T able XV for transit at P . T . 0 . the deflections will be
For“
sta . 410 0.
410 x .04 0°
68 .
41 1 x . 16 0°
4 12 x . 36 1°
12 .
4 13 x .64 1°
413 x 1 2°
Having set out the transition-curve, move to P . at sta. 413
backsigh t to and deflec t I ,°
2 8°
30'
2°
50 5°
and run out the circular curve to the
which suppose to fal l at sta. 420. Set the transi t at this point , andcause the vernier to read zero when the telescope is in tangent tocircular curve. T hedeflections taken from T ableXV wil l now be:
For sta. 420 68 , x .56 1°
35-2'
u u 421 36. X 2 2°
422 04 , x 4°
u 422 72, x 4°
a u 423 40, (60°
)o X 2 5°
128 A FIELD -MAN UAL FOR RAILROAD EN GIN EERS.
Set transi t at 423 40, the P . T . ; backsigh t to 420 and deflec t8°
30'
5°
40’
2° when the telescope wil l be in tangent .
155. F orm of T ransit N otes .-The fol lowing will i ll ustrate
a form of notes that will be found to answer .Let the P . 0 . of a 4
°
curve be at sta. 160 50 and a 200-ft.
transition~ curve be employed. Let the intersection-angle I beBy T able XVI , F ft .
, L°
a," z 100 ft . , so that
P . T . 0 . is at 159 50. T ake four 50-ft . stat ions on transi tioncurve and determine the defiec tion-angles as in the last section.
Deflec Calcu Map;tion (231m
l?! lated netic Rem arks .
angle.
gCourse. Course .
+50 O 12° 0'
5° 0 ’
3° 0’
1° 0’
+50 0 C .L . 4 ° 0' Set ver. at 2°
0° 45’ B S . to 159+ 50 , and0° 20
’
deflec t t o Run0° 5’ c ircular curve.
+50 9 P . T .C. 0° 0’ I1 200 . F
20 — 2 X 4
the central angle was W i th transi t at 161 50 set the
vernier to 2 ° backsight to 159 50,and deflec t into tangent
w i th the vernier reading zero . W i th the transi t at 164 50
cause the vernier to read zero when the transit i s in the tangent toc ircular curve, and run the last transit ion~ curve by deflec t ionsfrom this tangent . Wi th the transit at 166+ 50 backs ight to164 50 and deflect 4
°
2°40
’
1° when the telescopewil l
be in tangent and the l ine m ay be cont inued.
1 30 A FIELD -MAN UAL FOR RAILROAD ENGIN EERS.
By (186)
E , sec 13°
15’
ft.
T able XVI gives 1 2 hence the c i rcular curve will cover26
°
30’
2 X 6°
14°
or stat ions , so that the numberof the P . T . l wil l be (2 x 96
1 57 . T angent D istanc e , Offsets Unequal .
In Fig. 71 , 0 ,N , and K do not lie in the same straight l ine.
FIG . 71.
Draw PS perpendicular to NB,PQ perpendicular to LE Let
— K Q,
T'
T + F' cosec I F cot
'
I °
— T + F’
cosec I — F cot I ; (18 8 )
( 18 9
T 2 <6" T F
’
cot I F cosec I . (190)
EXAMPLE — Two tangents intersec t at sta. 820 and are to b (
united by a 6°
curve havingF c F’
and I : 31°
By T able IX ,T ft .
By T ab le XVI , l , 9 00, l, 260, x’
100, 27"
TRAN SIT ION -CURVES.
By
T. 100 x cosec 31 ° 48 ’ cot 31°
By
T 2 cot 31°
48' cosec 31 ° 48 2 400 5.
1 58 . T o Insert T ransition-curv es without C hanging the
P osit ion of the V ert ex,B .
In Fig. 72, ABC i s the located curve, FGHK the curve after
FIG . 72.
inserting transi t ion-curve. T he radius of the c ircular portion hasbeen changed from R to R
’in order to make room for the offset
PS F . BM : E i s the ex ternal to located curve, BL E'
the external to ci rcular curve having radius R '
and central angleI . In the t riangle LNM, LM LN see 41 : F see hence
E’z E — F see gl .
E m ay be found by (24) or by means of T able IX ; then E ’
becomes known, and from the same tableD’ is found by dividingthe tabular E by E ’
. D’
will be larger than D.
I t i s sometimes more convenient to assumeD'
and calculate E '
in the same manner as E ; then, from
F = (E'
( 192)
If this val ue ofF is too large or too smal l for the condi tions ofthe problem , a new D
’can be assumed and F recom put-ed.
1 32 A FIELD -MAN UAL FOR RAILROAD ENGIN EERS.
EXAMPLE .
— T he P . 0 . of 3. 5°
curve i s at sta. 182 , and angleI Compute the data for a new curve to al low for at ransition-curve with ft . offset .From T able IX, E 1 for I 40
° thereforeW F sec 20
°
x
then, by (191)E
’
5°
say 55
7
By T able XV I , for z, z 200,D z 5
.
7,
F : 2
For lx 220
F :
T hen for F : D 5°
7’
AD x
200 2002 5 8 .By 1 1
°
T he central angle for circular portion of curve is 40 — 2 X29 . equ ivalent to feet around curve.
In Fig. 72, B i s at sta . 186on the 5°
curve, and are BG
ft . on the 5°
7'
cu rve. T he P . 0 . 1 i s at 186 sta. 183
the P . T . 0 . at sta. 18 1 the
P . T . at 188 and the P . T . ; at 190
Had 17 been assumed equal to 5°
6'
or 51°
to begin wi th ,
we should have had E" z 36” then,by
F : X .93969 ft .
l, m ay be found by interpolation from T able XVI as above.
1 34 A FIELD -MAN UAL FOR RAILROAD EN GIN EERS.
R — R’
R — R"
T his is the same as (69) in 1 2 2 . I being known, set the transi t
at 0 , run out the curve ON , and inser t transi t ion-curve in the
usual way .
If I'
had been assumed in the beginning, R' could be found
fromSECOND METHOD .
— When the circular curve is flat , and shorttransit ion-curves are employed,we m ay compound the transit ion
curve with the c i rcular at the P . taking care that the difi erence of curvatures is not greater than 1 ° orAssume the posit ion of the P .0 . 1 from 100 to 200 feet from
the P . 0 . ; measure the perpendicular let fall from the P . 0 . ,
upon the tangent at the P . 0 . produced ; this w il l be T he
central angle I , can be calculated. knowing the length of
c ircular curve from the P . 0 . t o the assumed P . or the
angle between tangents m ay be measured wi th the transi t .T he coefiicients 0 and E of (140) and (142) m ay be foundfrom T ableXIV with I , q) as argument then,
from (140) and(142)
$3 1 = lj (1 — E ) .
Measure back from the foot of the perpendicular let fal lfrom the P . a di stance 2 , along tangent , and set the P . T .
Interm ediate points can be located, if needed, by 0 11t
from tangent , computed by (140) or thus at the m id
point the offset is §yhT HIRD METHOD .
— From formula ( 170)
0058 181 1
and from (36) y ,
T herefore8 70 71 21)
15071 21)
nearly .
. 0058 18Lo
1 1
TRANSIT ION -CURVES
But nD hence
1 : 150n ; ( 197 )
and as 10071 i s the length of c i rcular curve from P . 0 . to P .
l , is once and a ltalf as grea t.
From ( 146)200n1) 4
z. 1505 1505 ED ( 198 )
From this equat ion i t i s seen that if the break in curvatures isl imi ted to this method i s admissible up to D independentof the length of transition-curve.
EXAMPLE .— A 4
° curve is to have t ransition-curves inserted at
each end; compu te the necessary data.
BY FIRST MET HOD — Assume a -ft . offset , and the curvature to be changed from 4 °
to 5° by compounding . In T able I
find R R'
then, bycos I 1 99494 cos 5
°
5
267
stat ions, and, by T able
XVI , I ,°
so that the P .O. , wil l fal l ft . back of the
P . while the P . 0 . will be moved forward.289 stat ions or ft . ; the P . T . 0 . being, by Table XVI , 100 ft .back of the new P. 0 . will fal l 100 ft . back of old
P . 0 . T he t ransit ion-curve m ay now be located in the usualmanner.BY SECOND METHOD .
— Assume the P . 0 . 1 to fal l 150 ft . fromthe P . 0 . ,
mak ing I ,°
X 4 From T able XV , 0
.03488 , and, by y, z x 4 ft .
By
T he length of 5° curve is
Now, by
1 36 A FIELD -MAN UAL FOR RAILROAD EN GINEERS.
from which D '
5° which differs less than 2
°
from D .
BY
561 .001 1 ) ft .
T o find the posi tion of P . T . 0 . wi th reference to the old P . 0 .
consider that the distance from P . to foot of perpendicular fromthe P . 0 . 1 i s half the chord for angle 2L and can be taken from
T able IX , being equal to X T hen
2 feet i s the distance from old P . 0 . back to P . T .O
BY T H IRD MET HOD .
— Assume the P . to be 150 ft . from the
old P . (1 ; then, by Z , 225 ft . , and, by the curvatureof transi tion-curve at the P . 0 . 1 i s gx 4
°
5°
giving almostthe same resul ts as by the second method. Had we taken the
P . 160 ft . from P . 0 . we should have had l l 240,D
'
561 by interpolation from T able XV I ; the length alongtangent from P .0 . to foot of perpendicular from P . 0 . 1 ft . ,
and therefore ft . as the distance from P . 0 .
to P . T . 0 .
161 . T o Insert T ransition-curv es at the E U. and P . 0 . 0 . of
31 C o m pound C urv e b y C hanging the C urv atures of the F irstBranch .
In Fig. 74 let ARV be the located curve compounding at B.
Two cases occur .FIRST CASE .
— Second branch ha lving shorter radius.
T he ofi set at P . C.C. must be to outs ide of located curves ; let itbe EB F
2in the figure. Let OP F be known or assu m ed.
D raw the tangent BG, and draw EH paral lel thereto. Let OE
be the changed c urve, and CQ paral lel to tangent AH . Angle Im ay be computed fro m the known station num bers of A and B,
or m ay be measu red on the ground. T he new tangent distance i sEQ BG
’GK HQ(or L 8 ) . From the righ t triangle GHK ,
GK 2 BK tan GHK : F2cot I .
Simi larly , LS LWcosec I F cosec I . T herefore
T‘
EQ T F2 cot I F cosec I . (199)
T can be found from T able IX or fo r m ula then T' i s
known from T he degree of new curve, D ’
,m ay now be
found by m eans of T able IX , or from T able I by fi rst finding
TRAN SIT ION -CURVES.
F and being now known, the transit ion-curves m ay be
located.
EXAMPLE .
-A. 6° curve and a 4
°
curve are uni ted by a tangent540 ft . long; EH for 6
° curve ft . ; CG for 4° curve 3 ft . ;
B i s at sta. 180, 0 at 185 40 . Find F and F'
By tan a z:.0139 tan 0°
B wil l be m oved forwardgz . 133 stas . ft . to sta.
180 and 0 wi ll be moved backwards 2 stas. or 204
ft . to 185 20.
By F 1: ft .
By F’
ft .
T hese values cal l for l, ft . for 6°
curve, and l.for 4
° curve.
REMARK .— It wil l frequently be found that thi s problem
allows the l ine to be thrown "
on better ground. Should the
ground require tangent to be sh ifted inward,the curves must
be sharpened by compounding to admit of the necessary offsets .163 . H av ing Run a T angent which F alls Outside 51 L oc ated
C urtie,t o F ind the Offset F for a T ransition-curve Unit ingthem .
In Fig. 76let the tangent beAR ; CE the located curve. Set
transi t at some point 0 , and bring telescope into tangent tocurve. Measu re 0B and move toB , where angle ABC must bemeasured ; or measure OH per
pendicular to AR ; then
Now EG : R vers a or it isthe m id-ordinate for twice a
, and
m ay b e found from T able IX ; m g , 76,thenF = CH — EO = OH — R vers a .
T he point E is found from C’ by the relat ionEU
T he t ransition-curve m ay now be located.
140 A FIELD -MAN UAL FOR RAILROAD EN GINEERS.
164 . Insert ing T rans i t ion-c urv es in Old T rack .— Sec t ions
1 59 and 1 60 afi ord the means of inser ting transit ion-curves,of
which 1 59 is theoretical ly the best , though from the am ount oftrack disturbed i t m ay b e better to em ploy 1 60 . Som eti m es them ethod of 1 62 m ay be employed to advantage when the connec ting tangent is short . For easing the curves at point of com
pounding, the m ethod of 161 m ay be made use oi .T he offsets must necessarily be smal l if the new track is re
qui red to occu py the old road-bed. I t m ay be profitab le to add
to the road-bedwhen snflicient offset cannot b e secured for sharpcurves , though ordinarily much good can be accom pl ished evenwhen the new track is restr ic ted to the old road-b ed.
Unless the theoretical P . 0 . , and P . T . have beenmarked by monuments it m ay be diffi cult to retrace the old
l ines . If there i s plenty of room,the terminal tangents m ay b e
prolonged to intersect ion and I m easured, after which the degreeof curve m ay be found by measuring around curve and by approxim ate measurem ents of t he tangent distances ; then one or.
two assumptions and com putat ions w il l general ly suffice.
In cuts and rough country the curve m ay be run out by settingtransi t in center of road-bed and measuring the defiection-anglesfor a few po ints around the curve.
After the transi t ion-curves have been inserted per m anent monuments should be placed at each end of t t
'
ansition-curve to guidethe trackman in keeping up the proper superelevat ion of oute'rai l .165. Remark s on T abular Interp o lat ions . —Thegeneral inter
polat ion formula gi ven in algebra i s
Z z a + pdl +p (p - 2)
d
P(p ”( Z9 2X1? 3 )
in which t i s any term ,a the first term taken, p the number of
term s from a to t , d, the hi st from a of the first o rder ofdifferences, d? the first of the second order ofdifferences , etc .
In ordinary l inear interpo lation all terms after the second are
neglected ;‘
in interpolat ing by second differences all after theth ird,
etc .
In T able XIV l inear interpolat ion wil l answer for O and ordi
142 A FI ELD —MANUAL FOR RAILROAD EN GI N EERS.
Again, supposey ; to b e wanted when I, z 430 . By the form ula
33 x 2
1 0 1 0
49, +
W<2 ° 1 )
CHAPTER V .
FROGS’
AND SWITCHES .
ART ICLE 14 . TURNOUT S.
A. Turnouts from Straight Lines .
166. A T urnou t is a track used in leaving the main l ine. A.
F rog is placed at the intersect ion ofmain and turnou t railsfia . T he G auge
-line i s taken as c oinciding with inside face ofrai l . In making m easu rements between t racks the distance between corresponding gauge- l ines i s what is wanted.
b. T he G auge of track is the distance between gauge-lines ofthe rail s of that track .
0 . T he P oint of Switch i s the point at which the turnou t curvebegins for a point sw itch (spl i t sw i tch ) this is at the head-block ,whi lewith a stub swi tch i t i s the l ength of the swi tch-rai l backof the head-block , which is at the toe of swi tch .
d. T he F rog-p oint i s at the intersec t ion of the gauge-l ines of
intersect ing rai ls , and l ies a few inches in front of the bluntpoint offrog as manufac tured.
T he angle formed by the intersec t ing gauge-l ines is the F rog
angle.
6. T he P rog-num b er, N , i s the rat io of the axial length to the
width of base offrog.
FIG . 77.
In Fig. 77,
144 A FI ELD -MAN UAL FOR RAILROA D EN GIN EERS.
Letting the frog-angle BA 0 be F ,the figure yields
1 70 1t 1 2 L :angF
15 MV
11:0 0 12 22l
f. T he L ead, l, i s the di stance from point of switch to point of
frog , measured along that main rail in which the frog i s placed.
In Fig. 78 , CB l.
g. T he Stub -lead , s. l. , i s thedi stance alongmain rail from frogpoint back to a point where the turnou t ra i l diverges from mainrai l an amount equal to the throw . In Fig. 78 , K B llength of switch -rai l .
T he T hrow,t, of switch-rai l i s the di stance the point of a
spl i t switch, or toe of stub sw i tch , i s moved in opening or c losing
the swi tch . A distance of from 5 to 5% inches i s needed to gi venecessary c learance for flanges.
15. T he P rog-distanc e
, f.d. , i s the length of the chord of ou terrai l of turnou t from the point of a spl it switch,or toe of stub
swi tch,to the point of frog.
167 . G iv en the F rog-num b er
, IV, and the G auge, 9 ,of a
T urnout from a S traigh t L ine,' to F ind the L ead
, l, andRadius,R,
of C enter L ine ofT urnout .
’
FIG . 78 .
In Fig. 78 , AO 9 , CE l, angle ABC 1}F.
From the figure, I g cot 1517.But , (206) cot 4F 2N .
146 A FIELD~MANUAL FOR RA I LROA D EN G IN EERS.
1 68 . G iv en R (or D ) and g,t o F ind N ,
'
l,and F.
From (208) andR 5730
291) V517
From (207) andl : 29N : 2g‘/ V291? 107
107
9 ME. 0
F m ay al so be found from tr iangle OB C, Fig. 78
169 . T o F ind the L engt h of Switch-rail, 8 , when the F rog
num b er, IV, the T hrow of Switch, t , and the G auge, g,
are
G iv en.
In Fig. 78 , by geometry,
Neglec t ing the HG in denom inator as smal l
In like manner XL
Wri ting AG AB : CE S,and t aking the mean of de
no'
minators
whence S 1/ 2RiON Vy t .
5730W "
t'
R1 1 m g.D
Int:1,
FROGS AN D swn cnns. 1
1 7 0 . G iv en the M ain F rog-num ber, IV, to F ind the N um
b er, N . and L ead,l, of C ro t ch-frog for a T urnout from Bo th
Sides ofStraight M ain T rack .
In triangle OOH, Fig. 79, re
m em bering that RR 4N °
1: 1f 4N °+ 1
°
111155 , byN 1 = % COt é F] . o (218)
cos 11171
Fm m the figure and
9R 1L; (219)l l t an é F l
2N l
= gN l
(220)
Equat in’
g these values of l , and so lving for IV, gives
If the 71, in denominator be neglected as small compared with
21V”, (221) becomes
0 .707M
If in (220) we neglec t the 3, under radical , there‘i'esults
l , gN V2 1 . 414gN = 0. 707l. (223)
T he distance between main and crotch frogs measured alongma in rai l i s
l
or, approximately,1 z. 2gN 1 .414gN z 0 .586gN :
148 A FIELD -MANUAL FOR RAI LROAD EN GINEERS.
1 7 1 . T o F ind the Radius, R,
of T urnout and L ead, l, of
C rot ch-frog in T erm s of the C rot ch-frog N um b er, N 1
From N 2 2N 12.
Inser t this in (208 ) and giving
R : 29 . 2N 1" 4gN 1
’,
o o o oREMARK .
— In general the frogs kept in stock by manufacturersdo not afford su itable combinat ions of nu m bers for double turnouts . For instance, the theoret ical number of c rotch-frog for anu m ber 8 main frog is, by (221 ) or N , andwe shouldbe com pel led to use a nu m ber 51} or 6 for the c rotch -frog; th iswould necessi tate a different rate of curvature from c rotch to
main frog than from head-block to crotch .
1 7 2 . G iv en the N um bers of M iddle F rog, N 1 and ofM ain
F rogs,N and N t o F ind the Radii R, from P oint of Sw it ch
t o C rot ch-frog, and R and R’
, from C ro tch t o M ain F rogs.
In Fig. 80 we have, by (226)0 11V: R1 -N 1
2,
and,by (227)
N ow ifF , F , and F'
are the an
gles of the frogs N . , N , and N'
,
the angle
COH z -" F —" é Fx ,
COt (5217 + £17 1 )
Since CG 139 , the triangle (JE Gy ields
GE 419 cot 1}(F + é Fx) (228)
But , by trigonometry1 tan é F . tan iF,
tan gF—f tan iF,
150 A FIELD -MAN UAL FOR RAILROAD ENGI N EERS.
o 9By RS |6
'
C S 7,
ft
a 10°
28' curve.
m m 60
22 )ft .
1 7 3 . G iv en the N um b er, N ,of the Two M ain F rogs and the
G auge, 9 , t o find the C rot ch-frog N um b er,N 1 it s L ead,11 and
the Radius, R1 of C urv e through C ro tch when the D oub le
T urnout is to Sam e Side ofStraight M ain T rack .0
In Fig. 8 1 the frogs at B and G are of the same number, andm ay be taken as fal l ing on the same straight l ine through thecenter Angle OIGO 90
°
OGL F, and the t riangle
0 0 ,G i s therefore isosceles ; hence
OIG 0 1 0 0A
whence
FIG. 81.
b the same reasoning as in 167 , whence
N 137 51
‘/R 1
.
2g 4g 8
FROGS AND SWIT CHES. 1
Neglecting theaunder radical and writing R 291V“2gives
.707N ,
V2
which is identical w ith (222) for turnouts to opposite sides . For
EC and EB, as in 167 , l. 2gN , and l 291V. Hence
OB z z (237 )
EXAMPLE .
— Find N 1 , R1 , and l l , where N : 9 and
g ft .
By R x 81 ft . , 3. 7°
27’ curve.
1 ft . , a' 14
°
56’ curve.
N . . 707 x 9GB z z. x ft .
REMARK .— It m ay now be seen that the proper combinat ion of
frogs for a double turnou t to opposite sides appl ies also wherethe turnouts are to same side of straight main l ine. A lso theyapply to tu rnouts from opposite sides of curved main l ine when itsradius is not less than that required by m ain frog for straight1 7 4 . G iven the Num b er of M ain P fogs, JV, and of C rotch
frog, N , t o F ind the Radius ofCurv e. b etween P rog-points of
a Doub le T urnout to Sam e Side ofStraight T rack .
Fm . 82.
In Fig. 82, OQG 2 R2 159 , and the chord (JG must be determined. T he frogs at B and G being of the sam e number,0 2 0 0 GOO]
2 F and F].
52 A FIELD -MANUAL FOR RA ILROAD EN GIN EERS.
Draw GH perpendicular to EB ; then in triangle BGHG’H 9 cos F.
Draw OQL perpendicular and GE paral lel to EB ; from t riangles 0 2GK and 0 2 017,
(R; éc os F, cos 2F ) XL GH 9 cos F,
g cos F
cos F l— cos 2F
'
From triangle OQCG, since UOQG 2F
CG 2(R2 3139 ) sin %(2F F 1 ) . (239)
whence R2 39
EXAMPLE .-Given N 8 , N 1 r:6, and g to locate theturnou t.
By R 608 ft . ; R. 342 ft .
By R. 39 ft .
By CG ft .
1 75. Given the P rog—num b er, N ,the G auge, 9 ,
andD istanc e
29, b etween C enters,t o Unite M ain L ine with a Parallel Siding
when the Reversing-point is at P rog-point .
FIG . 53.
In Fig. 739 and BE are requ ired.
In triangleB0 ,E, B0 ,
R, 59 , E0 , R
1 + 139angle BO,
E F . By trigonometry,
154 A FIELD-MAN UAL FOR RAILROAD EN GIN EERS.
or, since cosec F N +4_
1
N’ (see
FIG . 84.
BK : (p — 29 )N +
EXAMPLE .
—For a No . 8 frog findB0 andBK whenp ft .
and g r: fi.
By B0 x 8 ft .
By EX 2 X 8 1: ft .
8 . Turnouts from Curves.
1 7 7 . G iv en the Radius of M ain C urv e,the P rog
-num b er,
and the G auge, t o F ind the Radius and L ead ofT urnout fromC onc av e Side ofM ain Line .
In Fig. 85, AB is the outer rail of tu rnout , CB the inner rai l ofmain track . In triangle OAB,
since OQBA OAB
OBA F,
OAB 180°
9,
and 0A 13 44 455 0 13 : R 49 .
FROGS AND SWITCHES. 1
T hen,by trigonometry ,
(R 119 ) (R 99 ) cot 46
(R 19 ) (R 49 ) tan tan
FIG . 85.
2B RR d t 46= — tan 1Fe ucm g co g 4 gN
T hen l r:B0 2(R 4g) sin 46.
If the length ofAB i s wanted,we can show that the angle
ABC 4F ; and by solv ing the tr iangle ABC, since A OB :
90° —M O
,
AB
T o findR, from triangle O,AB,
2(R2 49 ) sin 4(F + 6) AR.
Or, in triangle BOQC,
(R. 39 ) tan 41180 ( 17 + co t 9 17+ 9)
(R. 49 ) (R3 4g) tan 4F tan 4F
156 A FIELD-MAN UAL FOR RAILROAD ENGIN EERS.
Reducing and solv ing for R, ,
_ 9 9 1 1
2 tan 4F 5cot 4F . cot 4 (F + (249)
But, from trigonometry1 tan 4F . tan 46
cot 4(F+ 6) cot (4F+tan 4F + tan 46
Sub st itu te this in (249) and writeCOt é—F Z 2M tan iF :
ZN,
and reduce then
For 291V wr i teR1 , the radius of turnou t from straigh t track ,
and neglect the 49 in numerator as smal l compared wi th R ; then
R R
Now write5730
R.
5730
1) D 1
and reduce, yielding(252)
Formula (252) affords an easy m ethod of finding the degree ofturnout curve, or, if preferred, the radius m ay be first found by
Draw OE to the m id-po int of 0B ; OE does not differ greatlyfrom DE or 0 0 so, ifwe wri te OE R 69 , there results
gNQJV
z= 2<R — 391R
291VR
(253)
T he last term is qui te sm al l, even in the most extreme caselikely to arise in prac tice ; for a turnout from a 6
°
curve with
158 A F IELD-MAN UAL FOR RA ILROAD EN GIN EERS.
We m ay now fo l low the same l ine of reasoning by which ( 251)was derived, or more si m ply by assu m ing the tangent of the
difference of two smal l angles equal to the difference of thei rtangents ; that i s , tan 40 2 tan gF tan %6
Now i t can be easi ly shown that tan §0 2 él—éZ-V; therefore9 1V 1 9
_
1_
V
Ra 2N R
1 1 1 1 1whenceR2 R R, R
’
from which R9
R R,
5730Write R2 R1
5439
, and solve for D 9 .
D 2 D
D 2 I D 1 D ,
in which is the degree of turnout from straight track .
EXAMPLE .
— T urnout from outside of a 4°
curve, .N 8 ,zBy R, 1: x 64 r: 608 ft . , a 9
°
26'
curve.
By D 2 9°
26’
4°
5°
for which RBy l x X 8 z 76ft .
From ( 255) we have, by invert ingz 0
1 31
m m , by
l : 2870 X sin 1°
31’
ft
a difference of only ft . from the val ue given by
1 7 9 . T o F ind T heoretic al L ength of Swi tch-rail when the
T urnout is from a C urv ed T rack .
A com mon tangent being drawn at the swi tch -point, we shal lhave, as in 169
, for offset from tangent to main curve,
FROGS AN D SWIT CHES. l
the offset from tangent to turnout is
When the turnou t i s fr‘om concave side ofmain line,I y 3;
therefore
whence
5730 5730Writing RD
R2
D 2
and reducm g,
S 107 ID D 1
°
When the turnou t i s from convex s ide ofmain line,
15+
whence S
from whichS : 107
In (262) and (264) D ; is the degree of turnou t from straigh ttrack , and, as these formu las are ident ical w i th i t is seen thatthe theoretical length of swi tch - rai l on turnou ts from curves isthe same as on turnouts from straight l ine.
EXAMPLE .— Find 8 when t N 8 , g
By (208 1 608 feet, for which D , 9°
A FIELD -MAN UAL FOR RAILROAD EN GINEERS.
By or
feet .1 8 0 . G iv en the D istance p b etween C enter Lines ofCurved
M ain Line and S ide T rack,the F rog
-angle,F (or N um b er, Nand G auge, 9 , t o F ind the Radius and C entral Angle ofC urv eb eyond P rog
-
p oint .
FIRST CASE — Turnoutfi'
om outside of m ain line.
In Fig. 87, 0 is the center of main curve, 0 ; the center 0
FIG . 87.
curvewhose radi us is required. In triangle B0 0,0 0 : R + p BO = R+ %9
By the same reasoning as in 1 7 7 ,
212 19 2B p1: tan 1 17 :p 9 2N<p 9 )
In triangle OOIB,OIB RI é g; then, by the law of sines ,
sin 6Rl
— é g o
Also,
162 A FIELD-MANUAL'
FOR RAILROAD ENGIN EERS.
sin 63 1 4- 11? si
_
n(6— R)(R (272)
BE 2(R. 19 ) sin t (e F ) . (273)
0 . The Stub Lead.
1 8 1 . When the frog-num bg
er exceeds seven,the length of
switch ~ rail required to give the necessary clearance at heel becomes greater than is al lowed in practice. T o overcome this .
ditficulty sl ightly more cnrvature i s gi ven the switch -rai l ; moreover the physical point of sw itch is necessari ly some di stance inadvance of the theoretical po int . T he distance from heel of
switch to point ofmain frog wil l then be the same as from headblock of stub switch to main-frog po int , and i s
‘ termed the StubL ead. If to this di stance the length of swi tch -rai l be added, we
get the distance from the head-block of a point swi tch ‘
to the
point ofmain frog , Which is the Short L ead requ ired in pract ice.
1 8 2 . G iv en the T hrow , t , the G auge, 9 ,and the P i'og-num b er,
IV, to F ind the Stub L ead, 8 . l.
In Fig. 89 , K B i s the stub lead required; GN =XL ,the throw.
FIG. 89.
From (207) l 0B
and from (215) S
From the figure,
or 21V
FROGS ANn SWITCHES. 16
Formula (274) m ay be einployed for turnouts from curves aswel las straigh t l ines , since it was shown that the formulas from whichit was derived m ay be employed evenWhen the curvature ofmaintrack is considerable.
Below is a table of values of (g V9? ) for some of the morecommon values ofg and t.
T ABLE OF VALUES OF 9 Vyt .
3 Feet Gauge. 4 Feet Inch Gauge. 4 Feet 9 Inch Gauge.
Throw. g Vgt . Throw. g Vg—
t . Throw. g Vb} .
Inches. Feet . Inches. Feet . Inches. Feet .3 5 5
3} 54 3 .239 54 3 .275
4 5} 54
EXAMPLE .— Find the stub lead for N 8 , g ft t : 5
inches .
From the table, 9 V5 ft
and, by 16x ft .
1 8 3 . T he T urnout T ab le on thenext pagegives the frog-angles,the radi us of center l ine of tu rnout from a straight track and i tsdegree, the theoretical lead, the theoret ical length of swi tch -rai lfor t z 5 inches and the stub lead for certain values of t . T he
frog-numbers given cover all the usual cases .Suppose it required to find the short lead for a No . 9 frog and
5- inch throw when the gauge i s 4 ft . 9 inches and the length of
swi tch -rail 18 feet . From the table the stub lead i s feet ;hence the short lead is 18 feet , as againstft . for the theoret ical lead.
Inspec tion of the table w il l show that it makes no very greatdifi
'
erence in the tabular quantit ies whether the gauge be takenas 4 feet 85inches or 4 feet 9 inches . However, the nu m ericaloefiic ients in the form ulas involving y are somewhat s impler forthe latter val ue.
A FIELD-MAN UAL FOR. RA ILROAD ENG IN EERS .
TURNOUT TABLE FOR STRAIGHT TRACK .
4 FEET 85é INCH GAUGE.
DeTheoret Stub -l ead for a. Throwgree
Fro FroTheo Turn
of10 3 1
N0g
An 1g ret ical out
T urn Switchg 6.
Lead. Radius . ra i l for0“
t 5In. 5In. In. 534 In.
4 FEET 9 INCH GAUGE .
Degree Theoret Stub —lead for a ThrowF30 3 g
rog; $ 3331
T3??
0 ° “g e .
Lead. Radius . rail forout .
t = 5In. 5In. 534 111 . 534 111 .
0 feet feet feet feet feet feet4 . 14 15 37 42
5 11 25 24 8
10 23 19 56
6 9 32 342 0 16 468 48 61 75 14 16
8 10 12 19
7 38 10 44
8 7 9 76 00 9 25
6 44 80 75 8 21
9 6 22 7 27 58 95954 6 2 6 4 1 62 “3
0 5 44 6 2
1 1 5 12 4 59
12 4 46 4 11
3 4 24 3 34 84 294 4 5 3 4
5 3 49 2 41
166 A FIELD -MAN UAL FOR RA ILROA D EN GINEERS.
When the turnou t i s from a curve com pu te M from and
the m id-ordinate for a rai l 30 ft . long on main curve bythen the m id-ordinate for turnou t rai l w ill be the sum or difference of these values according as the turnout i s from concave orconvex s ide ofmain curve
ART ICLE 15. CROSSOVERS.
1 86. T o L ocate a C rossov er b etween P arallel StraightT rack s when the F rog
-num ber,the D ist anc e
, 17 , b etw een C en
t ers,and the G auge are giv en, insert ing ia T angent b etween
P rog-points.
FIG . 90.
In Fig. 90 i t is requ ired to find GB K 0 2, ME and
In the triangle BPM, BM : p 9 ; thenBE k BP EP
k = ( p —g) cosec F —
g cot F,
MK : (p g) cot F g cosec F .
From triangle OBO’ ofFig. 78 ,
0 0 R — 3g zR — gOB R + 3g 2B + g
°
FROGS AND SWITCHES.
In (a) write R 29117 2 by giving
4gN2 —
g 4N 2 — 1COSF —
4gN2 —l— g 4N 3
+ 1°
From Fig 78, triangle 03 0 ,CB 1 21
“ E "
013
"
R+ 3g 2R+ g'
Wri t ing 1 2gN and R 291Wgives4gN 4N
S’nF _
4gN2
+ g 4N 2 1‘
From trigonometry , tak ing theabove values of sinF and cosF,
1 1cosec F
sinF 41V"
cosF 1COt F
SinF E. o o o 0 o
Inserting theseval ues in (280) andk = ( p
MK = ( p — 2g)N
By GB XC z 291V; therefore
NO: 2z+ ME : 4gN + (p 29W4g,
Por NU
4N— l+ p
EXAMP LE .
— Find k andME for a No. 8 frog when p 13 ft .
and g ft .
By k x 3 T r . feet . ’
By ME x 3 feet.
168 A FIELD -MAN UAL FOR RA I LROAD EN GIN EERS.
1 8 7 . T o L ay Out a C rossov er in the F orm of a Reversed
C urv e.
Whenp is large, or for other reasons it i s desi rable to get awayfrom main track m ore rapidly than by the foregoing method, wem ay lay out the crossover in the form of a reversed curve.
FIG. 91 .
In Fig. 91 i t is required to fina GB HE and LE .
Find GB HE l by and the radius 0 0T hen, from we have
ME QR sin a .
The angle a i s gi ven by T hen
LH 2R sin a 2l. (285)
1 8 8 . T o L ay Out a C rossov er when a F ix ed L ength of T an
gent m ust b e Interposed b etween P oints of Rev ersal of
C urvature .
From the given frog-nu m ber determine the radius by (208 )then the p roblem m ay be sol ved by 1 32 .
1 8 9 . T o L ay Out a C rossov er in the F orm of a Reversed
C urv e when the T rack s t o b e Joined are C urv ed.
.In Fig . 92 let the no tation b e as shown. Let GH : R,
0 1M = R] , 0 2P = 0 20 : R2 .
0 0 9 2 R + p
1 70 A F IELD-MANUAL FOR RAILROAD EN GIN EERS.
ART ICLE 16. CROSSING-FROGS AND CROSSING -SLIPS .
A. Crossing-frogs.
1 9 0 . When two tracks intersect each other feur crossing-fi'
ogs
are required at the intersection of the two sets of rail s . T he fourfrogs are sometimes cal led a set of crossing-frogs.
1 9 1 . T o F ind the L ength ofRails Intercep ted b etween two
Intersec ting Straight T rack s whenthe Angle of Int ersec tion and the
T wo Gauges are giv en.
In Fig. 93, from triangle ABHAB E 0 g cosec F (288 )
and from triangle AEG,
AE B0 g, cosec F . (289)FIG . 93.
1 9 2 . G iv en the Angle of Intersec tion, a ,m ade by the C enter
L ines ofa St raight and C urv ed T rack , the G auges g:and 9 ,F ind the A ngles of the Set ofC rossing-frogs .
In Fig. 94, from the triangles OBK and OAH,
(R+ é g) cos F = R cos a+ §gu
R cos a1
cos F+ 5
In l ike manner,R COS a %9 1
cos F 1 :R —l-
lg
"
2"
.RCOS a £0 1
FIG . 94.
From triangle B0 0 to find the chord EU.
B0 9 03 3 ) sin 3041 F ) (294)
FROGS AND SWITCHES.
Sim ilarly ,GE : 2(R 4g) sin 4(Fg F3 ) .
From triangles ROM and COL , we have
E 0 ML (R 49 ) sin F , (R 4g) sin F2 . (296)
In l ike manner,
GB NR (R 49) sin F (R 4g) sin F 3 . (297)
1 9 3 . G iven the Angle of Intersec tion,a , m ade by the C enter
L ines of Two C urv ed T rack s.their G auges, g and 9 1 ,t o F ind
the Angles of the C rossing-frogs.
In Fig. 95, 0A R, R
and angle 0 11 0 , a of the triangleGAO, are given; whence 0 0 , m ay
be determined.
In triangle OBOI the side 0BR 39 , 0 13 : R 119 1 , and
0 0 1 z k are known, from which0 , we can determine the angle OBO,
Fm . 95.F .
In l ike manner from the triangle 0 00 1 determine F , , and
from triangle GEO, findF 2 . F 3 m ay be foundfrom triangle 0 0 0 1 .T o find the chord GB first find angle
B0 , 0 from triangle B0 , 0 , and angle0 0 1 0 from triangle GOI O; then
GB 2(R, 49 1 ) sin 40 0 13 (298)
In l ike manner ,E 0 2(RI 4g.) sin 4E0 , 0 , (299)
BC 2(R 49 ) sin 48 0 0 , ( 300)
GE 2(R 4g) sin 4GOE . (301)
When the tracks intersec t, as in Fig,
96, the sol ut ion i s evidently similar to Fm . 96.
the foregoing ,
1 72 A FIELD ~ MA N UA L FOR RAILROAD EN GIN EERS.
B. Crossing- s/lps .
1 9 4 . A C rossing-slip is an arrangem ent of sw i tch -rai l s in
connec tion w i th a set of c rossing-frogs , to connec t two t racksintersec t ing at a smal l angle.
1 95. G iv en the A ngle of Int ersec tion of Two StraightT rack s
, to F ind the L engt h and Radii ofCurv ature ofSlip -rails .
In Fig. 97 determ ine EA and
AR by 1 9 1 ; then assum e GE or EH
(ac cording as EA is less or greaterthan AR) as s m al l as the c rossingfrogs w i l l permit . Draw the radi iHO and G0 ; AH : A G k i s theknown tangent for the central angleF . Hence
0 0 13+ 39AH cot 4F : 9 76117, (302)
F m . 97
For the theoret ical length of rai l sz 1 l 304‘GB ><F
1LM (R 72 9 ) X
1 96. G iv en the Angle of Intersec t ion made by the C enterL ines of 3 Straigh t and a C urv ed T rack
,t o F ind the Radii and
L ength of Slip-rails .
FIRST CASE .—Slz
'
p -7'
ails inside m ain curve.
In Fig. 98 determ ine the angles F and F 1 at B and b 1 9 2 .
T hen assume K 0 as smal l as const ruc t ive reasons will perm i t .
1K Usm
1 K OO5
R + 39
5 BOK (F, F ) K OO, (307)
174 A F IELD-MAN UAL FOR RAILROA D EN GIN EERS.
Deter m ine angles 0 0 0 1 , and side 0 0 , by'
1 9 3 . MakeEM 2 K O) then
(R (E 1‘ 1‘ 59)
LOO —f 0 0 0
FIG . 99.
triangle MOO, two s ides and the inc luded angleknown
,and the triangle m ay be solved. 0 2 i s the center of
sl ip- rai l curves .
0 22110 1 MOO1 MO1 0
and M0 3 0 1 180 20 2510 1 .
From the isosceles tr iangle in which 0 1 111 and the threeangles are known
2 sin 41110 2 0 1
T hen
R, 49 R. 49 M0 2 , (315)
R2 z R, 4g M0 2 . (316)
T he central angle K OQE MO2 0 ; being known, GE and K L
m ay be found as in 1 96.
FROGS AN I) SWIT C H ES.
SECOND CAsm .-8 { z
‘
p-m ils on convex side of curves.
Let the dotted l ines of Fig. 99 represent th is case. AssumeA0 and compute angle AOQ produce 0 0 to the center ofsl ip - rai l curve make Og
’
N Reasoning as before, find
0 2'
N after which 0 2 38 , 0 2'
P , and the lengths ofTS and
QP m ay be found as in‘
the fi rst case.
Should the curves intersec t as in Fig. 96, no difficulty wil l befound in compu t ing the radi i and length of sl ip - rai l s by fol lowing the m ethods used above.
T hese methods furnish the theoretica l length of sl ip - rai ls ; bu tas the theoretical and physical swi tch -points do not coincide, theac tual length w il l be considerab ly less .
CHAPTER VI.
ART ICLE 1 7 . DEFIN IT ION S ; GENERAL CON SIDERAT ION S ; VERT ICAL CURVE S ; SUPERELEVAT ION OF OUT ER RAIL .
1 9 8 . T he work of locating the center l ine having been com
p leted, the field corps is usual ly di sbanded and a new one organized. T he C hief E ngineer st i l l remains in charge,
direct ing thework of construct ion,
passing on bids and estimates , arranging:contrac ts, and attending to such matters of im portance as his assistants are unprepared or unau thor ized to sett le.
1 9 9 . A D iv ision E ngineer is placed in charge ofa considerablelength of line, made up of several residenc ies . T o him the res ident engineers make reports , and from him receive di rect ionsand orders relating to construc tion. T hese reports wil l inc ludemonthly estimates, which are forwarded to the chiefengineer forinspec tion and approval . Pay -rol ls for the m en employed are
made out in the othee of the division engineer, and forwarded tothe chief.
2 0 0 . A Resident E ngineer i s placedin charge of a few miles ofline, called a Residency , and has direc t charge of the construotion.
’
He should have at least two assistants — a rodm an and an
axeman— and i t will be true economy to al low him also an
assistant who can take h is place at the instrument and assist insuperintending construc tion.
T he resident engineer i s usual ly requi red to set slope-stakes ,locate t restles and o ther bridges, tunnel s , culverts , c rossings, ando ther features preceding track- laying, and to m ake all measurements upon which esti m ates are based in determining the com
pensat ion of the contrac tor .Many roads prefer , espec ially on maintenance ofway,
to transpose the ter m s used above, so that the di v ision engineers report tot he resident engineer, whose residency m ay embrace severaldiv isions .
178 A FIELD -MANUAL FOR RAILROA D EN GIN EERS.
T he notes are recorded,however, in order that the contents
m ay be correctly calculated2 0 4 . A G rade-point is a point on the intersec tion of the p lane
of the road-bed with the ground-surface. If the ground i s levelt ransversely, a single stake at the center, marked will suffi ceto locate the point of passage from out to fill. When the groundi s no t level transversely , the l ine of intersectionwill be oblique tothe axis of the road and three grade-stakes areneeded, one at thecenter and one at each side.
If thewidth of road-bed in excavat ion differs from thewidth inembankment, the stake should be set at the edge of the widé stbase.
205. T o F ind the G rade-point when the Ground Slopes
Uniform ly b etween Stations .
FIG . 100 .
In Fig. 100 1et AB be the ground-l ine, F 0 the grade-l ine, and
E the grade-point . T he horizontal distance, x, from A to E is
requi red. Let the cut at A be h. the fill at B , 712 and the lengthof prismoid l.
,
Draw BG paral lel to CF. From the similar tri~angles ABE and ABG
IL! ha. 0 0 0 0 O 0 Q
If the ground does not slope uniformly , the point E m ust befound by trial , ’ such that the rod-reading equal s the differencebetween height
'
of instrument and elevat ion ofgrade.
206. V ert ic al C urv es.— T he angle formed by the junc tion of
two grade-l ines should be rounded off ei ther by substi tut ingseveral smal l changes for the one large one,or, preferably, by in
CF
CON STRUCT ION . 1
serting a regular curve. Where the a lgebraic dzjfi‘
erence of gradients i s less than no curve will be needed, while for largerdifi erences the length of vertical cu rve should vary with thatdifference, unless the c i rcumstances of the case— such as the
proximi ty of other ver t ical curves , or a bridge— should prescribei ts length . In any case the length m ay be either assumed, or a
gi ven rate of'
change per stat ion fixed upon and the length compuled.
The parabola i s espec ial]y wel l adapted for vert ical curves, because of the ease with which any correc tion m ay be found whenone i s known, since, as will presently be shown, the correct ionsvary as the square of the distance from the point of tangency .
A second property of th is curve enables us readi ly to find the
correc t ion at the vertex , or meet ing-point ofgrade-l ines . .
FIG. 101
In Fig. 101 let AOand 0B be the intersecting grade- l ines, andAFB the curve substi tuted for them . P roduce A O to E to
meet a vertical through B. Draw the vert ical CG. T hen willCF FG m by the second property referred to . Sincemeasurem ents are made hor izontal l y, the similar figures AUGand AEB furni sh the relation GE : 40 0 = 4EB. Cal l ing the
a lgebraic dtference ofgradients d,and the length of curve 21,
If the rate ofchange ofgradient per stat ion be a, it is evidentthat
T he equation of the parabola referred to A as origin m ay be
wri tten772
y '
l?(Qlw 0 o
1 8 0 A F I ELD-MANUAL '
Fon RA ILROA D ENG IN EERS.
T o find the co rrec tionEX 2 a t a distance a:from A , wehavefrom the s im ilar triangles AHL and A UG
But XL y, and 2 HL K L ; or, inserting val ues
2771513 27713 m a"
1 l 12
z m
g. 0
Insert the val ue of d from (0) in (a ) and the resul tant value ofm in then
12 15 x2 a
X2 l‘ 2
When a: 1 station, 2 . z 111 :when a; z 2 stat ions, 2 2 2a , etc .
I t wil l only be necessary to figure corrections for one-half thecurve, as they are the sam e for corresponding po ints each side of
the vertex . If preferred,however , al l correc tions m ay be com
puted fro m the first tangent produced.
EXAMPLE .
— A ~ l meets a grade at sta . 181 , the ele
vation of which is ft . Requi red the correc tions , and cor
rec ted grade elevat ions for points 100 ft . apart .Here the algebraic difference of gradients is
Suppose at be taken as or the length of curve as 6stations .Fo rm ula ( a) gives m 4 x 3 X feet .A t the P . 0 . , sta. 178 , z z 0 ; at 1 79 , (3 18 ) or gives 2 ,
at 180, 2 2 4 X .50 . T he original and correc tedgrade elevations are as fol lows
173 179 130 131 132 133 134
Origina l e leva tion. 89 2
Correc tions 0 0
Correc ted eleva t ’n 89 875 89 20
If a c irc le b e taken as the jo ining c urve we m ay derive by finding
R in term s of a ,then writ ing D 5730 R,
and n x,in form ula
EXAMPLE .— What wil l be the value of 71. when 6 .46, the
base being 14 feet ?By It x 7 x feet. T he outside i s thismuch higher than the center , the inside edge this much lower .T he superelevation of ou ter rai l should be com puted for thehighest speed at which trains are to be run over the curve; themaximum al lowed in pract i ce rarely exceeds 8 inches , since a
greater elevat ion would endanger the slow-running freigh t t rains .Even when the theoretical superelevation i s given the outer rail ,i t is more worn than the inner one, either because there are otherforces ac ting,
or because of the sliding ac t ion of the ou ter wheeldue to imperfect adjustment where the original coning has beendestroyed by wear.
Engineers sometimes elevate the outer rai l 1 inch per degree upto and make a z 34 inches for a 4
° curve, 4 inches for a 5°curve, and 44 inches for a 6
° curve. St i l l other rules are in use.
If transi tion-curves are not employed, the difference of elevat ion is the same from P . 0 . to P . T . , fading out to noth ing ontangent. T he elevat ion begins on tangent from 50 to 200 feetback of P . 0 . , depending on the amount the ou ter rail is to berai sed.
2 0 8 . E asing G rades on C urv es .— To compensate for the
increased resi stance due to curvature, i t is customary to reducethe grade on curves . T his resistance is taken to vary di rec tly asthe curvature; a rule often used is to reduce the gradientfoot per degree of curve
ART ICLE 18 . EARTHWORK.
A. Set ting Slope-s takes
20 9 . Slope-stak es are set at the points where the side slopesmeet the ground-surface, to mark the li m i ts of the excavat ion or
embankment,and to show the construc tor what the cut or fi ll
must be. In Fig. 102, KAE represents the ground-surface, HBOthe grade-su rface. Let AR h be the center height . Let
HL
XL
material ; for earth -excavation the side slope wil l average about 1to 1 , so tha t 3 z 1
,while for ordinary earth -em bankm ent it will
3 b e the side slope, which var ies with the nature of the
1 86 A F I ELD -MAN L’
AL FOR RAILROAD EN GIN EERS.
reading further out will be less, giving a correspondinglys m al ler d. o. ,
we try a reading at feet out . Suppose the reading to be the fi llwil l be cal l ing for a distanceout of feet , which agrees almost exac tly with the t rial distance. T he stake is marked F. and the result recordedin the c ross-section book .
On the other side of the sect ion suppose we est imate the fal l tobe feet in 15;weshould t ry a reading at 1 .5Xsay feet . Let th is reading be the fill wil l be
feet , cal l ing for a d. o. 7 X which showsour reading was taken too far out . T ry a reading at whichsuppose the fill is and the d.o. 7 +X 15. 4 , which agrees exac tly w ith the trial distance.
In excavation the method of proceeding i s the same as in em
bank m ent , except that 3 has general ly a different val ue. For sol idrock 3 is usual ly 4, that is, the slope is taken as 4 to 1 ; for looserock , gravel , and ordinary earth the slope m ay be taken as 1 to 1 .
T he stat ion constant in cuts i s always posi tive,and the rodreading has to be subtracted from it to obtain the out . In fi lls
,
when the HJ . i s grea ter than the grade height , the fill equal s thedifference of the rod reading and the station constant . Whenthe H I . i s less than the grade height the rod reading plus the
gives the fill.
2 1 1 . T he N o tes m ay be kep t in the form below, which represents one page of the c ross-sect ion book . T he cut or fill is writtenabove the l ine. the distance out below . A plus sign indicates acut , a minus sign a fill
S ta . Ground. Grade. Right.161
162
20 132 2 3;
43
66
163 185°1 1-6
CON STRU CT ION . 1 8 7
2 1 2 . Irregular Sec t ions— VVhen readings are taken only at
the center and sides i t is termed a th ree- level sec tion.
”Very
i rregular ground m ay require several more readings in o rder todetermine i ts area in th is case a reading is taken at each changeof su rface in the sec tion,
and the cut or fill, together with the distance out recorded— the distance being measured from the centerto the now t where the rod was held in tak ing the reading.
When the base cuts the ground- surface the sect ion i s partly inexcavation and partly in em bankm ent, but each side will bestaked out in the manner desc ribed above. T he distance of
grade-point from center must be found and recorded.
2 1 3 . St aking Out Openings .
— Where openings are to be leftfor trestles, cul verts , and other structures, stakes must be set to
mark the l imits of the embankment . Stakes marked T . B. are
set at the center and sides to fix the place where the top of bankis to end other stakes , marked F . S. ,
are set at the foot of slope,
the plus at which they fal l — together w i th the distance out fromcenter— being recorded in the note book. T he slope of the toe
ofdump should be the same as the side slope.
2 1 4 . M arking Stak es . — All slope and toe stakes that l imi texcavat ion or embankment should be dr iven with tops incl inedou tward from the center . T he cut or fill i s marked on inside inplain figures p receded by the letter C . or F . as being moreeasi ly understood by the contrac tor than the pl us and minussigns used in the notes . The reverse side should bear the stationnumber.
2 15. Shrink age— G xowth.
— It must be rememberedthat earthwork in embankment w i l l settle, or shrink in vol ume, even afterhaving been com pacted by the feet ofthe team'
s during construct ion. Where the fill i s not great , allowance m ay be made forshrinkage when set ting grade-stakes, but in heavy fills al lowanceshould be made when the stakes are set for construct ion. T he
proper al lowance wil l vary wi th the nature of the material, bu tabout 10 per cent wil l be a fai r average. T he contrac t shouldalways specify the amount of shrinkage to be al lowed on par
t icular works . If the earth is measured in the borrow -pi ts,an
equ ivalent al lowance should be made, since earth is more compac t in em bank m ent than before excavating.W i th rock , however. it is found that the vol ume increases
trapezoids whose area is positive, and one triangle whose area isnegative and equal to 4 hn(dn b) .
Writing out the area, we hav
'
e
761 )d1 (h; 11.2 )(d2 ( Z 1 )dn _ 1) hn(d,,
Perform ing the indicated operat ions and s impl ifying,
AR 20 o dn — lkn) ,
which is the same resul t obtained inEvidently 71 m ay have any posi tive integral value.
Ifpreferred, the c ross-sections m ay be plotted on cross-sect ionpaper and the area read off by means of a planimeter .2 2 1 . T ab les of A reas of L ev el Sec tions
,and the T hree
lev el C orrec tion .—Formula (322) m ay be employed in com
put ing the areas of level sections for any val ues of b and 3 .
T able XVII gives the areas for a few of these val ues. Whenmany sections are to be figured it wil l be wel l for the engineer tocompute the necessary tables, prov ided he is unable to securepubl ished ones for the particular bases and Slopes he is workingwith . It i s not wi thin the scope
,of th is volume to give the
variety of tables needed; they are publ ished elsewhere.
T he area of three-level sections m ay be found from the areas oflevel sections by the aid of a sui table correc t ion. Let the heigh tused in entering the tables of level sec t ions be the mean heigh t of
H 1 201 0the three-level section,
hm the correspondingarea, by is
A'
hm (2b hm s) 2bhm hm’s.
T he true area i s given by (321)A l hl
hob hoH l
H ) 2
4
710 761 271 0 Il l
2bhm 2kohm s ho2s.
3 71 028
From equat ions (a) and (b) the correc tion i sa z A’
A (hm? 2hohm (km (325)
1 94 A F IELD -MANUAL FOR RA ILROAD ENGIN EERS.
Adding (a ) , (b) , and the to tal volume isV z 7) +6 ’D
"
”0
6_ A l
a m + a m
l
+ a m 2 A.
“2 ‘ i” ( 1 2,
‘ 1" a s"
2 14 2°
therefore V : (A 1 i zim A 2) _
6’
the same asStated in words there resul ts the followingRULE .
-To the 371-771 of the end areas addfom' tim es the 'm t
'
d-area ,
m ultip ly by the length, and divide by 6. The result will be the
bolum e.
T o reduce to cubic yards , div ide by 27 .
Formula (327) contains three term s, the middle area beingderived from the c ross-sec t ion notes for the end sec tions at theexpense of so m e l i ttle trouble. In the attem p t to si m pl ify th isfor m ula Dr. George Bruce Hal sted in 188 1 publ ished a two - termpris m o idal formula , giv ing the Vol ume in terms of one base and
a sec tion at two thi rds of the length of the prismoid, the form ulabeing
(323)
In 1894 ProfessorW. H . Echol s showed by the aid of highermathemat ics that an indefinite nu m ber of two -term for m ulaemight be derived. T he same resul ts were establ ished in 1895byP rofessor T . U . T aylo r by elem entary mathemat ics .None of these two-ternzi fo r m ulae have so far been placed in a
fo r m sui table for appl ication to earthwork measurem ent, owingto the difficulty of finding the area of the auxi l iary sec t ion.
In fac t the only objec tion t o the use of (327) i s the loss of timerequi red in obtaining the m id-area and the uncertainty as to itsaccuracy in the case ofvery i rregular sect ions .For three- level ground we m ay construc t a section having
heights that are means between co rresponding end heights , butfor very i rregular sec tions there m ay b e uncer tainty as to wh atheights must be averaged to ob tain the m id -sec t ion heights . For
any other than the m id-sec tions the heights are obtained withmore difficulty.
CON STRUCT ION . 1 95
2 25. F orm of N ot es .
— T he record of areas and volumes m ay
be kep t in the form below , which represents the cross -sectionbook ,with the necessary columns added.
Sta . Ground Grade. L ' C ' R' £32918; £12223. 071
3
23675
+I30 .64
+2 4
180 0
179 0
If the method of averaging end areas is em ployed,the column
of m id-areas’
will not be needed, and m ay even be om i ttedwhencomput ing by the pris m o idal formula. In this case the notes form id-sec tion and the m id-area should be wri tten in red ink.
Ah othee record should be kep t in addition to the record in thec ross-sec tion book, to which it wil l not be necessary to t ransferthe elevations ofground and grade. If preferred, the areas andvol umes m ay be kept only in the othee record, omitting them inthe cross -section book .
2 26. P rism oidal C orrec tion.
— T he t ime and labor requ ired toobtain the area of the m id-sec tion m akes the use of the prismoidalformula objec tionable ; for th is reason the method of averagingend areas i s m os t often em ployed. The difference in the two
methods wil l no t be great , provided the difference in end heightsi s not over 3 or 4 ft . ; i t should never exceed 5ft .When the difference exceeds this a considerable error i s introdaood by t he use of I t wil l general ly be sufiic ient to
average end areas and then apply a correction if the resul t mustbe free from large errors .
(a ) C o rrec tion for L ev el Sec tions.— Between two level end
sec tions the vo lum e i s m ade up of one prism , one wedge, and two
pyram ids. For the pris m and wedge the true vol um e i s given by
CON STRUCT ION .
as wel l as transverse slope. Whatever method is employed, the
excavations and embankments must be separately computed.
2 2 8 . T ab les of V olum es for L ev el Sec t ions and E qual E ndAreas m ay be used in m aking prel i m inary esti m ates . T he average center heigh t for one or more stations is taken from the p ro
fi le and the vo lume at once read 0 11 fro m tables , such as T ableXIX
T able XX m ay be used in finding the volume, after hav ingaveiaged the end areas, and a co rrec t ion made by 2 26 ifdesi red.
2 2 9 . S ide D itches in cuts have a constant c ross - section, and
hence a constant volume for each ful l station. T hei r contentsare separately co m puted and added after the other computat ionshave been made. T hey need no t be shown in c rosso sec t ion notes .
2 30 . E ar thwork on C urv es . — In com puting quanti ties on
urves the end sec tions are assumed to be parallel , and the axia ldistance between sec t ions taken as the length of the prismo id.
If the vol u m e be taken as generated by a moving sect ion, and the
eenter ofgia vity of th is sec tion lie always on a vertical l ine pass ingthrough theaxis , this m ethod gives correc t resul ts otherwise not .The resul t w il l be too smal l or too large acco rding as the center ofgravi ty fall s w ithout or with in the center l ine of curve.
If thevol umes are com puted by averaging end areas , i t wil l beA useless refinem ent to apply a curvature correc tion:but if theprism o idal for m ula is employed, and accuracy is des i red,
itshould be appl ied, especial ly if the work be in rock .
T o find the curvature correct ion consider Fig. 107 whichrepresents the m ean sect ion of the pn
’
sm oid.
200 A FI ELD -MANUAL FOR RAILROAD EN GINEERS.
T he portion ABIIEG has i ts center of grav i ty on the l ine BF( BH having the sam e slope as BA ) hence the path of i ts centerof gravi ty wil l be t he sam e length as the axis of the prism o id,
and there will be no error in the computed volume generated bythis portion. In the triangle BCH draw BK to the mid po int ofOH . T he center of gravi ty of this triangle i s at M, two thi rds ofthe distance BK from B . N ow , by Guldin’
s rule ( theorem of
Pappus) the volume generated equal s the area m ul tiplied by thepath of the center of gravi ty , the center of rotation being in the
plane of the area.
Draw BL hori zontal and takeN on a vert ical through M let
the angle in degrees at the center be 6.
T he vol ume generated by the tr iangle BOB i sV B (R BN ) .
But the cal culated vol ume is776
°
180R'
Hence the curvature correc tion wil l be776
°
180V V0 z BCH RN .
_
2 2 d.+ d. d.+ d.
a o.
510130533. d.)0°
.006BCH (d1 + d3 )6°
. (332)
When the sections are 100 ft . apar t 6° D and the correc t ionbecom es
.006B CH (d1 dg)_D.
T he area of the t riangle BCH is easily seen to beA 761 ) 71. 0 (d2 (333)
If the triangle BCH is on the coiivex side of curve the correc
t ion must be added,if on the concave s ide i t must b e subtrac ted.
For l ight work the correc t ion is s m all , but for heavy work wi thsteep tranéverse m pe on sharp curves i t m ay be considerable.
In prac tice we m ay use the middle for the mean area withoutmaterial error .
EXAM PLE .
— Find the correc tion per station on an 8°
curve, 28
202 A FI ELD -MAN UAL FOR RAILROA D EN GINEERS.
300 ft and the price paid for overhaul 1 cent per c ubic yardper 100 ft .
T he addit ional co m pensat ion above the contrac t price wil l bew x 5000 x .01
ART ICLE 19 . GRADE AND BALLAST ST AKES, CULVERT S,BRIDGES, AND T UNNELS.
2 32 . G rade and C ent er Stak es .
— After the excavat ions andembankments have been brought approximately to the levelcal led for on the c ross - sec tion stakes , the engineer must gocareful ly over the road,
setting center stakes every hundred feeton tangents and flat curves , and every 50 or 25 feet on sharpcurves— the di stance between center stakes depending on the
sharpness of the curves. On tangents it will be suflicient to drivea grade-stake beside each center stake, so that its top w i l l be at
t he heigh t to which the finished surface must co m e, due al lowance being made for shrinkage.
On c urves grade-stakes must be set at each side a distanceequal to the half-base fro m the center ; the proper elevat ion or
depression of these stakes must be found by 2 0 7 ,form ula
T he P . C.
’
s and P . T .
’
s are recovered by means of the referencepoints set during location.
2 3 3 . Ballast -st ak es are set on the completed sub -
grade at theproper width of bal last -base— j ust as in slope- staking— with theirtops at the level of the final grade. T hey should be set at interval s of 50 ft . on tangents and flat c urves, and at 25ft . on sharpcurves .
2 34 . T rack C enters are set for the guidance of trackmen as
soon as the road-bed i s ready to receive the c ross - ties and rai l s .
2 35. T he Op ening left for a culvert , drain, or trestle bridge i sm easured from top of bank to top of bank ; the manner inWhichit should be staked out i s desc ribed in 2 1 3 .
A note of the si ze ofdrain and the m aterial ofwhich i t is to bebuil t , whether glazed earthenware pipe, b ox drain, stone culvert ,etc . , should be made in the note book opposi te the notes for theopening.
A fter the culvert or drain has been buil t the earth is filled in
CON STRUCT ION . 20
over and around it , and face or wing wal ls bu il t to protect thebank at the points where cul ver t or drain meets i ts face.
For trestle bridges it must be remem bered that the bank-si l l sset back from the top of bank a distance suffic ient to give firmbearing, usual ly about 6 ft . for o rdinary earth , and al lowancemade therefor in staking out the Opening. T he length of Opening is designated by the number of bents between bank-si l lsthus a 12-bent opening, where the distance between bents i s 14ft . , would be 13 x 14 12 170 ft . T he bent spacing dependsUpon the size of t imbers avai lable and upon the weigh t of locom ot i ves to be run over the road.
Whatever the nature of the structure , amplewaterway'
should
always be provided for the heav ies t storms ; fai lu re to do th is isf;he cause ofmany a costly wreck .
Center stakes are set for each trestle-bent , and ifpiles are to bedriven a stake should mark the posi tion of each pile. If the
bridge i s not at righ t angles to the stream it wil l often be best toset the bents askew, bu t this should be avo ided whenever possible.
After the piles have been driven cut -off levels are given by theengineer, for which a tack is set in the pile at a definite distancebelow the,
point of cu t-ofi , al lowance being made for cap ,
stringer , et c . If the bridge i s on a grade, the rate of r ise per bentmust he figured out. and al lowed for. On curves the propersuperelevat ion of ou ter rai l must be computed by the method of
20 7 .
For details of trest les see Foster ’s TrestleBridges.
2 36. The Piers and Abut m ents for truss bridges must be veryaccurately located, the spacing being done with a steel tapewhose constants are known, and the center and l imi ts beingmarked by stakes. Ou tangents the centers are easi ly locatedand referenced, bu t on curves this is not so easy, as the center oftrack cannot be taken as the center of pier on account of theclearance necessary for trains .Bridges on curves should be avoided whenever possible, butwhen they cannot be avoided the centers of piers are to be p lacedat the intersect ion ofpier-axis and bridge-chord.
”
In Fig. 109 ABC i s the center l ine of track , AE and CF the
pier-axes. A t the m id-point of the arc A O the tangent EF,
paral lel to AO, i s drawn; makeAN : N E CL LF , and drawN E
, which i s the bridge-ehord. T he points N and L are the
centers of the piers
CONSTRUCT ION 2
2 4 1 . A Progress P rofi le should accompany the monthly estimate to exhibi t graphically the amount ofwork done dur ing themonth , different colors being used for the different months . T he
final profile should show approxi m ately the progress of the work .
The colors m ay be laid on with a brush , or hatchings made wi th apen; in nei ther case should the color obscure the l ines oi- the profile-paper. A dupl icate progress profile should be retained in thedivision engineer ’s othee; if transparent profile-paper i s employed,
one m ay be simply traced through from the other. A fur theradvantage of the transparent paper i s that blue-prints of any portion of the profile m ay be readi ly made when duplicates are
desired, provided the drawings are in black or any color admi tt ing blue printing.
2 4 2 . M asonry i s to be measured in cubic yards, and any
material on hand, bu t not in place, i s to be measured and est imated. T he classification
_oi masonry must be according to
specifications. Foundat ion-pi ts for piers or cul verts must bem easured as soon as comp leted, and before the masonry has beenput in place.
2 4 3 . Bridges must be est i m ated by measurement , or bycheck ing up material in place and that on hand but not in place.
For trestle bridges, or foundations requi ring pil ing, the actualnumber of l inear feet below cap mus t be measu red; th is necessitates the constant supervision of the engineer or an assistant
,
someti m es known as a“ pile- recorder, ” whose duty i t is to seethat all piles come up to specifications and are driven in accord
ance therewi th .
All framing-t imber in place,or del ivered bu t no t in place, i s
to be inc luded in the estimate, the amount being obtained bymeasurement .Steel spans or trestles are to be estimated, in the same manner
as wooden trestles , by checking up or m easuring the material onhand and in place.
2 4 4 . T rack M aterial must be checkedup ei ther by the“‘
m aterial clerk or the engineer in charge of t rack . Bal lasting properly belongs w ith the graduation , but m ay be pu t in place afterthe rails have been laid; in ei ther case it i s est imated in accordance with the specificat ions.
For prel iminary and monthly est imates it wil l be sufficient to
208 A FIELD -MAN UAL FOR RAILROA D E N G I N EERS.
estimate track m aterial by means of tables showing the num berof eross - t ies for a given spac ing and the weigh t of steel for a
given rail sec tion, but before the final estimate is m ade all mater ial must be m easured or counted.
2 4 5. B lank E s t i m ate-sheets are sent out from the ch ief engim eer ’s ofiice to be fi lled out by the engineers m aking estimate,
who should retain a copy of each est im ate rendered . On thesesheets should appear the total quant i ty est i m ated
,the amount of
theilast preceding estimate, and the esti m ate for the month ,which
will be the difference of the other two .
T he division engineer’ s esti m ate must show not only the quan
t i ty of m ater ial , but i ts val ue in dol lars and cents com puted fromthe contract price. T he foot ings of the several col um ns thenserve as a check upon each other .2 4 6. T he M onth ly P ayments are not m ade for the ful l
amount estimated but about 15or 20 per cent i s retained unt i lafter the final estimate has been made,
in order to insure thecompletion of the work by the contractor
, and to be used as a
fund from which to withhold the am ount of damages provided inthe contrac t for fai lure to comply w i th all i ts provisions .
E x tra s inc ident to minor changes , or to the protection or
drainage of thework , are usually shown on the final esti m ate, but
a better way would be to requi re the cont rac tor to present h is bil lfor extras at the end of each m onth , and to
.
incorporate them in
the monthly est imate when they are j ust . -T he engineer shouldtake measu rem ents upon any extra work at the tim e of i ts comp let ion , and should keep a record thereof. If the ex tras are of a
nature no t admitting of measurem ent , he should note the com
pensation to b e al lowed at the ti m e the extra work is done.
24 8 . T he Final E st imate must include all earthwork moved,all material in bridges , all masonry in foundat ions, culverts, piers ,and tunnels , and all other material supplied or work done incompliance with the contrac t . T he engineer should keep hisno tes ful l and complete during the construc tion of the work , in
o rder to be able to meet the contractor ’s c lai m s for extras or com e
plaints as to classification. Any i tem s that m ay have been overlooked in making up the m onthly est i m ates must b e includeo.
here.
TABLE I.— RADII.
Deg . Radius.l Deg . Radius. Deg. Radius . Deg . Radius . Deg.
00 O’ Irninite 1
°
343775.
17188 7 .
4911o. 7 l
38 197 2 I
2644 4 2 1
0'
2° 0' 3
° 0'
1 1 1
2 2 2
3 3 34 4 44
5 5 5
6 6 67 7 7
8 8 8
9 9 9
10 10 10
11 1 1 1 1
12 12 12
13 3 13
14 14 14
15 15 15
16 4523 44 16 16
17 17 17
18 18 18
19 19 19
20 20 20
4°
173648
O/
co
oo
-ra
owm
oo
wu
HO
wHH
HHH
t-AHt-J
u
o
co
oo
-t
cu
ona-co
wv-A
TABLE II. —MINUTES IN DECIMALS OF A DEGREE . 215
. 01667
.06667
. 11667
. 16667
.21667
. 23333
.26667
. 31667
. 41667
. 45000
.46667
.51667
.61667
. 71667
. 75000
. 81667
. 86667
. 91667
. 96667
10”
. 01944
.0361 1
. 0861 1
. 1027
. 11944
. 13611
. 18611
. 21944
. 2361 1
. 25278
. 26944
.28611
. 30278
. 31944
.38611
.4 1944
.43611
.46944
. 4861 1
.51944
.5361 1
.5861 1
.6027
.65278
.68611
. 71944
. 73611
. 8 1944
.8361 1
. 91944
. 03611
.95278
. 96944
. 98611
10”
15'
. 004 17
. 03750
A N
.U l
. 10417
. 13750
. 154 17
. 18750
.237505417
. 30417
. 33750
. 35417
. 38 750
. 404 17
. 43750
. 45417
. 47083
.554 17
.57083
.58750
.60417
.62083
.63750
.654 177083
.68750
. 70417
. 73750
.854 17
. 90417
. 92083
. 95417
15"
. 03889
.08889
. 13889
. 15556
. 18889
7222
.30556
. 32222
. 35556
. 40556
.45556
.50556
.53889
.55556
.58889
.60556
.62222
.65556
. 93889
.95556
30”
.04167
.24 167
.34 167
. 44167
.59167
. 01111
.61111
.62778
. 71111
40”
.52917
.62917 .
45”
.01389 0. 02917 .03055 1
.04722 2.06111 . 06389 3.07778 .07917 .08056 4
.09 167 5
. 10833 . 11111 . 11250 . 1 1389 6. 12917 . 13056 7
. 14167 . 14444 . 14722 8. 16111 . 16389 9
. 17500 . 1777 . 18056 10
. 19167 . 194 44 . 19583 . 19722 11
.21 1 11 .21250 . 21389 12
.22917 . 23056 13
14
.26111 .26250 15
. 27500 .2 7778 . 27917 .28056 16
. 29167 .29444 .29583 .29722 17.31111 .31250 .31389 18. 3277 . 33056 19
20
.36111 . 36389 21. 37500 . 37778 .37917 22
. 39167 . 39444 .39583 23
. 4 11 1 1 .4 1389 24
. 42778 . 42917 .43056 25
.44583 26
. 46111 .46250 27
. 47778 .47917 28. 49167 . 49444 . 49583 29
.51111 .5138 9 30
313233
.58056 3435
.61339 3637
.64 167 .64722 38
.65833 .66111 39
.67500 .67778 .67917 40
.69167 .69583 41
.71389 42
. 72917 43
.74583 44
. 76111 . 76389 45
. 77778 .77917 . 78056 46
.79167 47
.80833 . 31111 .8 1250 48
. 82778 . 82917 49
.84 167 50
. 85833 86111 . 86389 51
. 87500 .37917 52
. 89167 . 89583 . 89722 53
.91111 . 91250 . 91389 54
.92778 . 92917 . 93056 55
. 94167 . 94722 50
. 96111 . 96250 57. 97500 . 97778 . 97917 58
99167 50
218 A F IELD -MAN UAL FOR RAILROAD EN GINEERS.
T ABLE IV .
— LONG CHORDS .
hDegree Ac tua l Arc ,
Long C O l ds
of one 1 2 3 4'
5CUFVG Station. Sta t ion. Sta t ions. Sta t ions . Sta t ions . Stat ions .
T ABLE V .
— MID-ORDINATES TO LONG CHORDS.
Degree of 1 2 3 4 5
Curve. St a t l on. Stat ions . Stat ions . Stat ions . Stat ions . Stat ions .
TABLES. 21
TABLE V .— MID-ORDINAT ES TO LONG CHORDS .
Degree of 1 2 3 4 5 6Curve. Stat ion. Sta t ions . Sta t ions . Sta t ions . Sta t ions. Stations.
1 24
1 0
1 2
2 69
TABLE V I.
— LOGARITHMS OF NUMBERS .
I
O{
3 T ABLE V I .
— L‘OGARITHMS OF NUMBERS.
'
23 £3 41 £5 ( 3 7'
25 S)
47712 4 7727 47741 4 775647770 47784 47799 478 13 47828 47842
78 71 78 85 790078578001
8 144
828 7
8430
8572
8 714
8 855
8996
8015
8 159
8302
8444
8586
8 728
8 869
9010
8 029
8 173
8 316
8458
8601
8 742
8 888
9024
3044
3 137
3330
3473
36153 756.83979 033
79 14
8058
32028344
8487
86298 770
8 911
9052
7929
8073
82 16
8359
850 1
8643
8 926
9066
7943
808 7
8230
8373
8515
86578 799
8 940
9080
7958 7972 7986
8 101
8244
8387
85308671
8 8 13
8 954
9094
8 1 16
82598401
8544
8686
8 827
89689108
8 130
8273
8416
8558
8 700
8 841
8982
9 122
4913649150 49164 8 49192 4920649220 49234 49248 492629276 9290 9304 9318 9332 9346 9360 9374
,9388 9402
9415 9429 9443 9457 9471 9485 9499 9513 9527 9541
9554 9568 9582 9596 9610 9624 9638'
9651 9665 9679
9693 9707 972 1 9734 974 8 9762'
9776 9790 9803 98 17
9831 9845 9859 9872 9886 9900 9914 9927 9941 99559969 9982 999650010 50024 50037 50051 5006550079 500925010650120 50133 0 147 0 161 0 174 0 18 8 0202 0215 0229
0243 0256 0284 02970379 0393 0406 0420 0433 0447 0461 0474 0488
5051550529 50542 5055650569 50583 5059650610 50623 506370651
0736
0920
10551 1331322
14551537
1720
0664
~
0799
0934
1068
1202
1335
1468
1601
1 733
0678
08 13
0947
108 1
1215
1348
148 1
1614
1746
069 1
0 326
0 961
10951223
1362
14951627
1 759
0705
0311_0325 0338 0352 0365
0 718 0732
0840 0853 08660974
1 108
1242
1375
1508
1640
1772
098 7
1 121
1255
138 8
1521
1654
1786
1001
1 1351268
1402
1534
1667
1 799
0745
0880
1014
1 148
1282
14 151548
168018 12
0759
0893
1028
1 162
1295
1428
1561
1693
1825
0501
0772
0907
104 1
1 1751308
1441
1574
1706
1 838
51851 5186551878 51891 51904 51917 51930 51 943 51957 519701933
2114,
2244
23752504
2634
2763
2392
3020
1996
2 127
2257
238 8
2517
2647
2776
29053033
2009
2140
2270
2401
2530
2660'
2789
2917
3046
2022
2 153
2284
24 14
2543
2673
2802
2930
3058
2035
2 166
2297
2427
2556
2636
23152943
3071
2048
2179
2310
2440
256926992827
2956
3084
2061
2 1922323
24532532
2 711
2340
2969
3097
2075.2205
23362466
2595.
27242353
29323 1 10
2033
22 13
2349
2479
2603~
2737
23662994
3122
2101
2231
2362
2492
2621
2750
28 79
3007
3 135
53148 53161 53173 5318653199 532 12 53224 53237 53250 532633275
3403
3529
3656
3782
3908
4033
4 158
428 3
3288 3301
34 15
3542
3668
37943920
’
4045
4 170
4295
3428
3555368 1
3807
3933
4058
4 183
4307
3314
344 1
3567
3694
3820
3945
4070
4 195
4320
33263453
.
3580
3706
3832
3958
4083
4208
4332
3339
3466
3593
3719
38453970
4095
42204345
3352
3479
3605
3 32
3 57
3983
4 108
4233
4357
3364
3491
3613
3744
3370
3995
4 120
4245
4370:
3377’
3390
3504? 35173631 3643
‘
3757. 31769
3882 52895
4008.4 020
4 133 4 145
4258 4 270
4382 4 .394
54407 5441954432 54444 5445654469 54481 54494 54506545318
226 TABLE VI .
— LOGARITHMS OF NUMBERS .
TABLE V I.
- L6G’
ARITHMS OF NUMBERS .
O l 2 3 4 5 6 7 8
227
9
6532 1 65331 65341 65350 65360 65369 65379 65389 65398 6540854 18 5427 5437
5514
5610
5706
5801
5896
5992
5523
5619
5715
58 1 1
5906
6001
5533
5629
57255820
5916
60 1 1
6087 6096 6106
618 1 619 1 6200
66276662856629566304 66314 66323 66332 66342 66351 663616370
6464
6558
6652
6745
68396932
6380
6474
65676661
67556848
694 1
6389
6483
65776671
6764
6857
6950
7025,
7034 7043
71 17 7127 7136
67210 67219 67228 67237 67247 672566726567274 67284 672937302
7394
74367573
7669
7761
7352
7943
3034
731 1
7403
7495
7587
7679
7770
7861
7952
8043
732 1
74 13
7504
7596
768 8
7779
78 70
7961
8052
5447
5543
5639
5734
5830
5925
6020
61 156210
6398
6492
6586
6680
6773
6867
6960
7052
7145
7330
7422
7514
7605
7697
7788
7879
7970
8061
5456
5552
5648
5744
5839
59356030
6124
6219
6408
6502
6596
6689
6783
6876
6969
7062
7154
7339
7431
7523
7614
7706
7797
78 88
7979
8070
5466
5562
5658
5753
5849
5944
6039
6134
6229
6417
651 1
660566996792
68856978
'
707 1
7164
7348
7440
7532
762477 15
7806
7897
7988
8079
5475557 1
5667
5763
53535954
6049
6143
6233
6427
6521
6614
6708
680 1
6894
698 7
7080
7173
7357
7449
754 1
7633
7724
78 15'
7906
7997
8088
5485
558 1
56775772
5868
5963
6058
6153
6247
64366530
6624
6717'
68 1 1
6904
6997
7089
5495559 1
5636
5732
5377
5973
6063
6162
6257
64456539
6633
67276820
6913
7006
7099
5504
5600
5696
5792
588 7
5982
60776172
6266
64556549
6642
6736
6829
6922
7015
7108
7182 7191 7201
7367
7459
7550
7642
7733
782579 16
8006
8097
7376
7468
7560
7651
7742
7834
79258015
8 106
73857477
7569
7660
7752
7843
7934
8024
8 1 15
68 124 68133 68 142 68151 68 160 68 169 68 178 68 187 68 1966820582158305839584858574
8664
8 753
8 842
8 931
3224
3314
3404
3494
3533
8673.
8 762
8 851
8940
8 233 8242
8323 8 332
8413
8502
8592
868 1
8 771
8 860
8949
8422
851 1
8601
8690
8 780
8869
8958
8251
8341
8431
8520
8610
8699
8 789
8878
8966
8260
8 3508440
8529
86198 708
8 797
8 8 868975
8269
8359
8449
8538
8628
8 71 7
8 806
8 895
8984
8278
83688458
8547
8637
8 726
8 8 15
8904
8993
8287
8377
84678556
8646
8 735
8824
8 913
9002
82968 386
8476
8565
86558 744
8 833
8 922
9011
69020 69028 69037 690466905569064 69073 69082 69090 69099
9 108'
9197
92859373
9461
9548
9636
9 117
9205
9294
938 1“
9469
9557
9644
9723 9732
98 10 98 19
9 126
9214
9302
9390
9478
9566
9653
9740
9827
9 1359223
931 1
9399
9487
9574
9662
9749
9836
9144
9232
9320
9408
9496
9583
9671
9758
9845
9 152
924 1
9329
94 17
9504
9592
9679
9767
9854
9 161
9249
9338
9425
9513
9601
9688
9775
9862
9 170
9258
9346
9434
9522
9609. 9697
9784
9871
9179
9267
9355
9443
9531
9618
9705
9793
9880
9 18 8
9276
9364
9452
9539
9627
9714
9801
9888
69897 6990669914 69923 69932 69940 69949 69958 6996669975
2 8 TABLE V L— LOGARITHMS 10 1? NUMBERS .
TABLE V I .
— LOGARITHMS OF NUMBERS .
60 778 15 77822 77830 77837 77844 77851 77859 7786677873 77880
6
9
O1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
01
2
3
4
5
6
7
8
78 8 7 7895 7902
7960
8032
8 104
8 176
8247
8 3 19
8390
8462
8604
8675
8 746
8 8 17
8 888
8958
9029
9099
9 169
9309
9379
9449
9518
95889657
9727
9796
9865
7967
8039
8 1 1 1
8 183
8254
8326
8 398
8469
861 18682
8 753
8 824
88958 965
9036
9 106
9 176
9316
9386
9456
9525
95959664
9734
9803
9872
7974
8046
8 1 18
8 190
8262
8333
84058476
8618
8689
8 760
8831
8 902
8972
9043
9 1 13
9 183
9323
9393
9463
9532
9602
9671
974 1
98 10
9879
7909
798 1
8053
8 125
8 197
8269
8340
84 12
8483
8625
8696
8 767
8 838
8 909
8 979
90509 120
9 190
9330
9400
9470
9539
9609
9678
9748
98 17
9886
79 16
7988
8061
8 132
8204
8 276
8347
84 19
90
8633
8 704
8 774
8845
8 916
8986
9057
9 127
9 197
9337
9407
9477
9546
96169685
9754
9824
9893
7924
7996
8068
8 140
821 1
8283
83558426
8497
8640
8 7 1 1
8 78 1
8 852
8923
8993
9064
9 134
9204
9344
9414
9484
9553
9623
9692
9761
9831
9900
793 1
8003
80758 147
82 19
8290
8362
8433
8504
8647
8 7 18
8 789
8 859
8 930
9000
9071
9 14 1
921 1
9351
9421
949 1
9560
9630
9699
9768
9837
9906
7938
8010
8082
8 154
8226
8297
8369
8440
8512
8654
8 7258 796
8 866
8937
9007
9078
9 148 \
9218
9358
9428
9498
9567
9637
9706
9775
9844
9913
7945
8017
8089
8 161
8233
83058 376
8447
8519
8661
8 732
8 803
8 873
8944
9014
90859 155
9225
9365
943595059574
9644
9713
9782
9851
9920
7952
80258097
8 168
8240
8 312
8383
84558526
78533 78540 78547 78554 78561 78569 7857678583 78590 785978668
8 739
8 8 10
8 880
8951
9021
9092
9 162
9232
79239 7924679253 79260 79267 79274 7928 1 79288 79295793029372
9442
9511
958 1
9650
9720
9789
9858
9927
6 0 79934 7994 1 79948 7995579962 79969 7997579982 79989 79996
1 80003 80010 800 17 80024 80030 80037 80044 80051 80058 800652
3
4
5
6
89
01
2
3
4
5
6
7
8
9
0
0072
0 140
0209
0277
0346
04 14
0482
0550
0079
0147
0216
0284
0353
0421
0489
0557
0085
0154
0223
029 1
0359
0428
0496
0564
0092
0161
0229
0298
0366
0434
0502
0570
0099
0168
0236
03050373
0441
0509
0577
0686 0693 0699 0706 07130754 0760 0767 0774 078 1
0821
0889
0956
1023
1090
1 158
1224
0828
0895
0963
1030
1097
1 164
0835 0841 0848
0902 0909 09 16
0969 0976 0983
1037 1043‘
1050
1 104 1 1 1 1 1 1 17
1 171 1 173 1 134
010601750243
0312
03300443
0516
0534
0720
0737
03550922
0990
0 113
0 132
0250
0313
938704550523
0591
0 120
0 18 8
0257
0325
0393
0462
0530
0598
0127
0 195
0264
0332
0400
0468
0536
0604
0726 0733 0740
0794 0801 0808
0862
0929 0936
0996 1003
1057 1064 1070
0868
1 124 1 131 1 137
1 191 1 198 1204
0875
0943
10 10
1077
1 144
12 11
1231 1238 1245 1251 1258 1265 1271 1278
0 134
0202
027 1
0339
0407
0475
0543
0611
80618 8062580632 80638 8064580652 80659 8066580672 806790747
08 14
0882
0949
1017
1084
1 151
1218
1285
8 1291 8 1298 8 13058 1311 8 1318 8 13258 1331 8 1338 8 13458 1351
670am
TABLE V I.
—LOGARITHMS OF NUMBERS .
O 1 2 3 4: 5 ( i 7 8
235
9
92942 92947 92952 92957 92962 92967 92973 92978 92983 9298 82993
3044
30953 146
3197
3247
3298
3349
3399
2998
3049
3100
3151
3202
3252
3303
3354
3404
3003
3054
3105
31563207
3253
3303
3359
3409
3008
3059
3110
3161
3212
3263
3313
3364
3414
3013
3064
3 115
3 166
3217
3268
3318
3369
3420
30 18
3069
3 120
3171
3222
3273
3323
3374
3425
3024
30753125
3176
3227
3278
3328
3379
3430
3029
3080
3131
318 1
3232
3283
3334
3384
3435
3034
30853 136
3186
3237
328 8
3339
3389
3440
3039
3090
3 14 1
3192
3242
3293
3344
3394
3445
93450 9345593460 9346593470 9347593480 9348593490 934953500
3551
3601
3651
3702
3752
3802
3852
3902
3505
3556
3606
3656
3707
3757
3807
3857
3907
3510
3561
361 1
3661
3712
3762
38 12
3862
3912
3515
3566
3616
3666
3717
3767
38 17
3867
3917
3520
3571
3621
3671
3722
3772
3822
3872
3922
3526
3576
3626
3676
3727
3777
3827
38 77
3927'
.3531
358 1
3631
3682
3732
3782
3832
3882
3932
3536
3586
3636
368 73737
378 7
3837
388 7
3937
3541
359 1
3641
3692
3742
3792
3842
3892
3942
3546
35963646
3697
3747
3797
3847
3897
3947
93952 93957 93962 93967 93972 93977 93982 9398 7 93992 93997
4002
4052
4 101
4 151
4201
4250
4300
4349
4399
4007
4057
4 106
4 1564206
4255
43054354
4404
4012
4062
4 1 1 1
4 161
4211
4260
43 10
4359
4409
4017
4067
4 1 16
4 166
4216
4265
43154364
4414
4022
4072
4 121
4 171
4221
4270
4320
4369
4419
4027
4077
4 126
4 176
4226
4275
43254374
4424
4032
4082'
4 131
4 18 1
4231
4280
4330
4379
4429
4037
4086
4136
4 186
4236
4285
43354384
4433
4042
4091
4 14 1
4 19 1
4240
4290
4340
4389
4438
4047
4096
4 146
4 196
4245
4295
43454394
4443
94448 94453 94458 94463 94468 94473 94478 94483 9448 8 94493
4498
4547
4596
4645
4694
47434792
4841
4 890
4503
4552
4601
46504699
4748
4797
4846
4895
4507
4557
4606
4655
4 704
4 753
4802
4851
4900
4512
4562
461 1
4660
4709
4758
4807
4 856
4905
4517
4567
4616
46654714
4763
4312
4361
4910
4522
4571
4621
4670
47 19
4768
48 17
4866
4915
4527
4576
4626
46754724
4773
4822
48 71
4919
4532
458 1
4630
4680
4729
4778
4 827
48 76
4924
4537
4586
4635
46854 734
4783
4832
48 80
4929
4542
459 1
46404689
4738
478 7
4836
48 85
4934
94939 94944 94949 94954 94959 94963 94968 94973 94978 949834988
5036
5085
5134
5182
523 1
5279
5328
5376
4993
5041
5090
5139
5137
5236
5234
5332
5331
4998
5046
50955143
5192
5240
5289
5337
5386
5002
5051
5100
5148
5197
5245 5250
5294 5299
5342
5007
5056
51055153
5202
5347
5012
5061
5109
5158
5207
5255
5303
53525390
'
5395 5400
95424 95429 95434 95439 95444 95448 95453 95458 95463 95468
5017
5066
51 14
5163
521 1
5260
5308
5357
5022
5071
51 19
5168
5216
52655313
5361
5027
5075
5124
5173
5221
5270
5318
5366
5405 54 10 54 15
5032
5080
5129
5177
5226
5274
5323
537 1~54 19
2361
17
TABLE VI .
—LOGARITHMS OF NUMBERS.
1 . 23 £3 4 . 55'
6 7'
£3'
5)
900 95424 95429 95434 95439 95444 95448 95453 95458 95463 954685477 5482 548 71
9
wwr-‘Q
(Q
CD
'Q
Q
UI
NP'
OO
IQ
5472
5521
5569
5617
56655713
5761
5809
5856
5952
5999
6047
60956142
6190
6237628 4
6332
6426
6473
6520
6567
6614
6661
6703
67556302
68 956942
0 6988
7035708 1
7 128
7 174
7220
7267
7359
7405
7451
7497
7543
7589
7635768 1
7727
5525
5574
5622
5670
5718
5766
58 13
5861
5957
6004
6052
6099
6147
6194
6242
6289
6336
6431
6478
65256572
6619
66666713
6759
6806
6900
6946
6993
7039
70867132
7179
72257271
7364
74 10
7456
7502
7548
7594
7640
7685
7731
5530
5578
5626
5674
5722
5770
58 18
5866
5961
6009
60576104
6152
6199
6246
6294
634 1
6435'
6483
6530
6577
6624
6670
6717
6764
68 1 1
6904
5535
5583
5631
5679
5727
5775
5823
5871
5966
6014
6061
6109
6156
6204
6251
6298
6346
6440
648 7
6534
658 1
6628
6675
6722
6769
68 16
6909
5492
5540
5588
5636
5684
5732
5780
5828
58 75
5971
60 19
6066
61 14
6161
6209
6256
6303
6350
64456492
6539
6586
6633
6680
6727
6774
6820
6914
5497
55455593
564 1
5689
5737
57855832
58 80
5976
6023
6071
61 18
6166
62 13
6261
6308
6355
64506497
6544
6591
6638
66856731
6778
6825
6918
6951 6956 6960 69656997
7044
7090
7137
7183
7230
7276
7368
74 14
7460
7506
7552
7598
7644
7690
7736
7002
7049
7095
7142
718 8
7234
7280
7373
74 19
7465751 1
7557
7603
7649
76957740
7007
7053
7100
7146
7192
7239
7285
7377-7424
74 70
7516
7562
7607
7653
7699
7745
7011“
7058
7104
7151
7 197
7243
7290
7382
7428
7474
7520
7566
7612
7658
7704
7749
5501
55505598
5646
5694
5742
5789
5837
5885
5980
6028
6076
6123
6171
62 186265
6313
6360
6454
6501
6548
6595
6642
6689
6736
6783
6830
6923
6970
7016
7063
7109
7 155
7202
7248
7294
7433
7479
75257571
7617
76637703
7754
5506
5554
5602
5650
5693
5746
5794
5342
5390
5985
6033
6080
6128
6175
6223
6270
6317
6365
6459
6506
6553
6600' 66476694
674 1
6733
6334
6923
6974
7021
7067
71 14
7 160
7206
7253
7299
7437
7483
7529
7575
7621
7667
77 13
7759
551 1
5559
5607
5655
5703
5751
5799
5847
5895
5990
6038
6085
6133
6180
6227
62756322
6369
6464
651 1
6553
66056652
6699
6745
6792
6339
6932
6979
70257072
71 13
7165721 1
7257
7304
7396
7442
7488
7534
7580
7626
7672
7717
7763
5516
5564
5612
56605708
5756
5804
5852
5899
95904 95909 95914 95918 95923 95928 95933 95938 95942 95947
59956042
6090
6137
61856232
6280
6327
6374
96379 96384 96388 96393 96398 96402 96407 96412 9641 7 964216468
6515
6562
6609
6656
6703
6750
6797
6844
96848 96853 96858 96862 96867 968 72 96876968 8 1 96886968906937
6934
7030
7077
7123
7 169
72 16
7262
7303
97313 97317 97322 97327 97331 97336 97340 9734597350 97354738 7 7391 7400
7447
7493
7539
7585763
7676
7722
7768
97772 97777 97782 97786 97791 97795978 00 97804 97809 978 13
TABLE V II.
— LOGARITHMIC SINES AND I
COSINES.239
9 7240 TABLE V IL— LOGARITHMIC S INES AND COSINES.
09101 9966
99655
15596
15770
15944
9953299727
99701 12799
99695
9968 1 13994
Cog ne
Sine38519
1.
F 1a
17
TABLE dIQQm GARIT fhflc SINES ANbQJOSINES . 243
15°
Sine
1 11 m
j .
1
244 TABLE V IQ§ LOGARPI% fiIC S INES AND
49844
51702
9 52705
53161
53370
Cosine
97558
97541
97532
7439
7408
97363
Sine
53578
53647
53922
54635
54702
54836
55169
Cos ine S ine
g 1 1 1 R 1 143
246 TABLEL 7 1 — LOGARI
’1‘11MIC S INES 111131 COSINES.
726°
Sine Cos ine0
1 96067
2 60988
3 61016 96056
4 61045 96050
5 96045
6 6110 1 96039
7 61 129 64365
8 61 158
9 61186
10
1 1 61242 9601 1
12 61270 96005
13 61298
14 95994
15 95988
16 61382 9598 2
17 61411 95977
18 61438 95971 6464719 61466 95965
20
2 1 61522 95954‘ 2 61550 95948
23 61578 95942 64775
24 61606 95937
25 6163 4 95931 64826
26 1 61662 95925
2728 61717 959 14 95192
29 61745
30
31 9589732 61828 95891
33 61856 958 8534 61883 95879 6505435 6191 136 61939 958687 95862
38 95856 636100
40 . 9584 44 1 62076 95839
42 9583343 62131 9582744 6215945 62186 958 1546 62214 958 1047 62241 9580448 9579849 62296 95792 95059
50 .6232351 62350 9578052 62377 9577553 62405 9576954 62432 9576355 62459 95757 95397
56 9575157 62513 95745 9538458 9573959 62568 95733 95372
60 62595 95728
Cos ine S ine S ine
S ine Cosine Sine S ine Cos ine
90072
79224
9 . 90043 9 79256
900 14
Cosine
256 TABLE V III .
—LOG . TANGEN T S AND COTANGENT S .
21102
218 14
2 197 1
22670
'
23435
23661
78898
7818678 107
77484
7656576490764 14
24926
26372
28323
30457
30975
32498
32623
Co t an
69674
69478
69348
69025
Tkzn
TAB"
TABLE V III.
— LOG . TANGENTS AND COTANGENTS.263
3 1 °
268 A FIELD-MANUAL FOR RAILROAD ENGIN EERS.
TABLE IX .
—FUNCT IONS OF A ONE -DEGREE CURVE.
The Long Chords , Mid-Ord inates,Externals, and T angent
D istances of th is table are for a curve of 5730 feet radius . T o
find the correspond ing fp nc tions of any other curve d iv ide thetabular values by the degree ofcurve.
For m et r ic curves hav ing 20-metre chords , m ul tip ly th e degreeby 5and enter the table w i th the resul t as a value ofD
, the tabular values being taken as m et res instead of feetT hus for 3. 1 °
30’
m et ric curve hav ing I : 45°
the tangent d ist ance i s T metres . Again,
suppose I 38°
and the long cho rd m . known and D required. T he
tabular L . C . is 3731 m . ; therefore I)
IX.— FUNCT IONS OF A ONE -DEGREE CURVE. 27 1.
13.
23 23
23 68
Tl
518 12
523 16
550 06
IL ( 1
1108 3
40 59
277c)N
12 CL
IL ( 1
1647 .
1677 .
DI .
43 .
43 .
44 .
44 .
44 .
45.
45.
46.
46.
47 .
59 .
57 .
57 .
58 .
74
98
22
59 19
59 92
IX .
—FUNCT IONS OF A ONE -DEGREE CURVE .
74928 7
w v.)
7s4f35
856 35
DI.
'
55.07
63 9364 18
52 13
52 36
53 51
54 6754 . 9 1
65 6765 0 3
773 .
re» .
~1
-1
-1
33
8
8
8
8
4
793 -39
798 . 49
80 1 . 89
887 m
888 . 7
890 . 44
895. 56
897 . 27
898 .98
902 . 40
905. 8 1
IX .
— FUNCT IONS OF A ONE -DEGREE CURVE.
IL CL 11 . 13.
IL ( 1
2843 5
150 2
157 5
-A03
00
03
40
36
omnocnwoo
fik
CD
OI
H'Q
WQD
CJU-‘Q
CO
CI
A
O
Q
CO
«1
-J
-1
-1
«1
-1
HHh—du—ly—AHH
p-lv-A
H
u—JHHH
'
H-A—AH
—A—n
u-au
co
co
oo
oo
R)
388
182 5
F"
‘
1
R1
-1
‘I
‘
Q
‘1
“Q
R?
H
-Ai—AH
HH
HHH
HHHM
“1
CD
m
wh
h
b
a
b
d
~
q
wm
a
.
n
UP.
1354 5
1358 0
GP.
148 1 9
2 7 °
275
[ 1 CL
2914 5
2966 1
61 . IE. 11
166 1
167 4
17 .617 . 0
17 . 4
BI . 13. 11
18 3 3
194 4
IX .- FUNCT IONS OF A ONE-DEGREE CURVE. 279
IL ( 1 BI . IE.
4 147 3
4162 8
11 ( 1
4296 1
38 ? 4
386 0
386 5387 2
398 0
Bl .
425 4
428 6
435 6
420 8
Tfl
'2249 4
2255 2
43 °
[ b (1
1“ CL
4462 44465 5
B4 . EL
3
7
439 2
408 5
409 8
4 12 2
4 15 3
437 8
31 . 13.
4766
446 4
449 5
450 8
453 4
Tfl
Tfl
2410 6
2428 3
WG
Q
MO
'
10
14
16
18
2022
24
262830
32
34
36
38
40
424 4
46
48
52
545658
I » ( L
4 77 .
91
459 . 4
461 . 3
467 . 9
468 .6
469 . 3
470 .6
471 . 2
471 9
472 .6
473 .2
91 .
501 .5
502 . 8
507 .6508 . 3
509 .6
IE.
549 .6
55239
563 . 7564 .5
565. 3
566. 2
Ti
2449 . 9
2469 .
2471 .
2473 .
2475.
2477 .
247
248 1 .
2483 .
2485.
248 7 .
2489 .
2491 . 01
0
01
0
01
0
03
03
03
05
'
Q
2551 .
2553 .
2555.HHy—L
2559 .
2561 .
2563 .
2565.
2567 .
2569 .
2571 .
73 .
2575.
2577 .
257 .
2581 .
2583 .
2585.
2587 . 2
2589 . 2
HHL
MH
L
Hp-AHHHU—‘p—tund
280 lX — FUNCT IONS OF A ONE -DEGREE CURVE .
4 7 °
4569 .
4572 . ~1
-4
IL ( L
DI . IE.
4 9 °
BI. 13. Tfi
282 IX.
— FUNCT IONS OF A ONE-DEGREE CURVE.
I4. Cl 91 . FL TP.
7
2974 4
645 9 728 0
647 4 729 9
91 . IE.
'
P.
7
7
77 . 8
13. (P.
730 9
745 7
749 .
750 .
751 .
752 .
753 .
754 .
—1
TflB
3 1 13:3 115.
3 117 .
3 1 19 .
3 131 .
3124 .
3 126.
3 130 .
3 132 .
3 134 .
3 137 .
3 139 .
3 141 .
3143 . O‘
fi
NJ
O
QO
-7cb
80718
820 .
IX .
— FUNCTIONS OF A ONE-DEGREE CURVE .283
IL ( 1 Bl . IE. flfl
BI , 13. 11
9 1517
In ( 1 Dl .
7
05
03
05
0?
Q
Q
Q
U!
QCD
O
‘Q
BI .
IE.
853 5
13.
1 2
328 1 6
11
53614
I“ CL
5930 9
5945.
5947 .
5950 .
5953 .
5956.
5959 .
5962 .
5965.
5967 .
5970 .
5976.
5979 .
5982 .
5985.
5987 . oo
o
v—‘oo
wx
Q
CD
O
NJ
OO
Ol
m
wQO
t-fl
IL ( L
10
k:
cncn
éo
99
QO
(I)
Na
‘T
U‘
UP
O
(D
O’b
v‘k
31 . IE.
1054 .
1055.
1056.
1057 .
1059 .
1060 . (O
O
‘J
U'
hD
O
'
3645
'
3634 .
11
3456.6
3167 .
77 0
11
3617 .
3620 .
3622 .
3624 .
3627 .
3629 .
363 1 .
"Q
WO
"
3636.
3638 .
’
3641 .
3643 .
3648 11
IX .
— FUNCT IONS OF A ONE-DEGREE CURVE .
IL ( L
60 19 - 0
IL ( L
6157 .
6160 . 3
DI .
845. 3
846. 2
847 . 9
848 . 8
855. 8
856. c
85 .6
858 . 4
859 . 3
860 . 2
862 -8
868 . 9
869 . 8870 7
DI .
IE.
990 .3
99 1 .5
992 . 7
1004 . 4
1008 4
1009 .6
1013 2
1024 . 2
1025. 4
1026 7
108 9 . 4
1090 .6
11
3552 . 7
3662 . 2
3669 . 2
286 IX .
- FUNCT IONS OF A ONE -DEGREE CURVE .
IL ( 1
IL ( L
6789 8.
DI .
1058 4
1094 3
1095 2
13.
13.
11
TE
4222 0
In ( 1 11 .
1 77 .
1 133 8
13 . 11
18
CD
CNQ
MO
’
IX .
— FUNCT IONS OF A ONE -DEGREE CURVE . 28 7
11 ( 1
IL ( 1
7 134 0
7 4°
DI . IE.
'
76°
11 .
1238 4
1239 51240 5
1243 6
13.
15465
1590 0
11
I1 ( 1
7 178 2
7204 2
11
1253 9
1255 0
13. 11
4609 .8
9[d 8 8 IX.
—FUNCT IONS OF A ONE-DEGREE CURVE .
79 °
1 1 ( 1
11 ( 1
7368 9
7417 1374 19 8
1803:
11 . IE. 11
11 .
HH
t—Ah—AH
H
-1
~J
OE
CD
OE
G
U'
UV
UY
O'
O‘
17
3
71
Nah
OD
-I
QO
O
AD
Q
U‘
Q
CD
O
1
1805.
11
4885 3
53
11 CL
I1 ( 1
1 1 .
1333 0
DI .
F4
0
0
2
8
8
8
O
WG
A
MO
QD
N
O'
WH
(
0
0
00
0:
172
b
um
m
q
wuwwq
wo
w
13. 11
'
4975. 1
290 IX .
—FUNCT IONS '
OF A ONE -DEGREE CURVE .
8 6°
7
7
BI . 13.
1
0
N.
)
00
03
‘
2
0'
10
10
05
00
5377 .
h fi fl
I“ CL 11 . 11
IX .
— FUNCT IONS OF A ONE -DEGREE CURVE .
IL CL
8 117 6
1 1 C1
DI . 13 .
13.
11
11
29 1
IJ. ( L
8376 8
838 1 3
m
v-tb-A
u-tu-nu-A
N
O
O
CD
-1
03
3UZO
-)Q
HHHHH
D—‘HHH
H
m
q
uwuc
q
m
wb
wq
m
a
wo
m
H ‘I(O
N)
G
OV
17
172
173
173
173
HN)
10
1-40
23
00
9 3 9
91 . 13. 11
6094 7
6144 7
IX .
— FUNCT IONS1 06°
IL ( L
In CL
DI . 13.
1 0 8 °
DI . 13.
4109 3
OF A ONE -DEGREE CURVE .
1 0 7°
Tfl
fll
7
8023 3
c)N 95
13.
3964 2
1 0 9 °
2443 4
Tfl
8 178 2
296 IX .
—FUNCTIONS OF A ONE -DEGREE CURVE .
1 1 0 °
4268 3
9393 2
2451 6
2459 8
1 1 2 °
8559 4
460624610 8
8635.
8640 .
8646.
8651 . O
t
~l
i
O
1 1 1 °
DI . 13. (P.
8378 9
8394 6
1 1 3°
DI . 13 . TP.
HHHHHH
H
l
Gb
OI
AOO
Mv-fi
Oto
m-QQUUA
OO
MHO
59
Sine Cos in. 08716. 08745
. 08774
.08803
. 08831
. 08860
.08889
.08918
. 08947
. 08976.09005
. 09034
. 09063
.09092
.09121
. 09150
. 09179
.09208
. 09237
. 09266
. 09295
.09324
. 09353
.09382
.09411
. 09440
.09469
. 09498
.09527
. 09556
.09585
.09614
. 09642
. 09671
. 09700
. 09729
. 09758
. 09787
.098 16
. 09845
.09874
109903. 09932
. 09961
. 09990
. 10019
. 10048
. 10077
. 10106
. 10135
. 10164
. 10192
. 10221
. 10250
. 10279
. 10308
. 10337
10366. 10395. 10424. 10453
Cosin S ine Cosin S ine Cos in S ine
. 99619
. 99617
. 99614
. 99612
. 99609
. 99607
. 99604
. 99602
. 99599
. 99596
.99594
.99591
. 99588
. 99586
. 99583
. 99580
. 99578
. 99575
. 99572
. 9957
. 99567
.99564
. 99562
. 99559
. 99556
. 99553
. 99551
. 99548
. 99545
. 99542
J99540
.99537
. 99534
. 99531
. 99528
. 99526
. 99523
. 99520
. 99517
. 99514
. 99511
.99508
. 99506
. 99503
. 99500
. 99497
. 99494
. 99491
. 99488
. 99485
. 99482
99479
. 99476
. 99473
. 99470
. 99467
. 99464
.99461
. 99458
. 99455
. 99452
TABLE X.
—SINES AND COSINES.
S ine Cosin Sine Cosin. 10453
. 10482
. 10511
. 10540
. 10569
. 10597
. 10626
. 10655
. 10684
. 10713
. 10742
. 10771
. 10800
. 10829
. 10858
. 10887
. 10916
. 10945
. 10973
. 1 1002
. 11031
. 11060
. 1 1089
. 1 1 118
. 1 1 147
. 1 1176
. 1 1205
. 1 1234
. 1 1263
. 1 129 1
. 11320
. 11349
. 1 1378
. 11407
. 11436
. 11465
. 1 1494
. 11523
. 1 1552
. 11580
. 1 1609
. 11638
. 1 1667
. 1 1696
. 1 1725
. 1 1754
. 1 1783
. 118 12
. 1 1840
. 1 1869
. 1 1898
. 11927
. 1 1956
. 1 1985
. 12014
. 12043
. 1207 1
. 12100
12158. 12187
. 99452
. 99449
.99446
. 99443
. 99MO9 9 37. 994349 9 8 1
. 9942 8
. 99424
.99421
. 99418
. 994 15
. 99412
. 99409
. 99406
. 99402
. 99399
. 99396
. 99393
. 99390
.99386
. 99383
. 99380
. 99377
. 99374
. 99370
.99367. 99364. 99360
. 99357
. 99354
. 99351
. 99347
. 99344
. 99341
. 99337
. 99334
. 99331
. 99327
. 99324
.99320
. 99317
. 99314
. 99310“
99307. 99303. 99300
. 99297
. 99293
. 99290
. 99286. 99283
. 99279
. 99276
. 99272
. 99269
. 99265
. 99258
. 99255
8 3 °
. 12187
. 12216
. 12245
. 1227
. 12302
. 12331
. 12360
. 12389
. 12418
. 12447
. 12476
. 12504
. 12533
. 12562
. 12591
. 12620
. 12649
. 12678
. 12706
. 12735
12764
. 12793
. 12822
. 12851
. 12880
. 12908
. 12937
. 12966
. 12995
. 13024
. 13053
. 13081
. 13110
. 13139
. 13168
. 13197
. 13226
. 13254
. 13283
. 13312
. 13341
. 13370
. 13399
. 13427
. 13456
. 13485
. 13514
. 13543
. 13572
. 13600
. 13629
. 13658
. 13687
. 13716. 1374 4
. 13773
. 13802
. 1383L
. 13860
. 13889 '
. 13917
. 99255
. 99251
.99248
. 99244
399240
. 99237
. 99233
. 99230
. 99226
. 99222
. 99219
. 99215
. 99211
. 99208
. 99204
. 99200
. 99197
. 99193
. 99189
. 99186
. 99182
. 99178
. 99175
. 99171
. 99167
. 99163
. 99160
. 99156
. 99152
. 99148
. 99144
.99141
. 99137
. 99133
. 99129
. 99125
. 99122
. 99118
. 99114
. 99110
.99106
. 99102
. 99098
. 99094
. 99091
. 99087
. 99083
. 99079
. 99075
. 99071
. 99067
. 99063
. 99059
. 99055
. 99051
. 99047
. 99043
. 99039
. 99035
. 99031
. 99027
82°
I 8 0
S ine Cosin. 13917. 13946. 13975
. 14004
. 14033
. 14061
. 14090
. 14119
. 14148
. 14177
. 14205
. 14234
. 14263
. 14292
. 14320
. 14349
. 14378
. 14407
. 14436
. 14464
. 14493
14522
14551
14580
146081463714666146951472314752
14781
148 1014838148671489614925
14954
14982150111504015069
150971512615155
1518415212
15241
1527
15299
15327
15356
15385154 1415442
15471
1550015529
155571558615615
. 15643
Cos in Sine
. 99027
. 99023
. 99019
. 99015
. 99011
. 99006
. 99002
. 98998
. 98994
. 98990
. 98986
. 98982
. 98978
. 98973
. 98969
. 98965
. 98961
. 98957
. 98953
. 98948
. 98944
98940
98936
989319892798923989199891498910
9890698902
98897988939888998884988809887698871988679886398858
98854
98849
98845
98841
98836
98832
98827
98823988 1898814
9880998805
98800
98796
98791987879878298778
98773
. 98769
S ine Cosin. 15643. 15672. 15701. 15730
. 1575815787158 16158451587315902
15931
1595915988
1601716046160741610316132161601618916218
16246162751630416333163611639016419164471647616505
16533165621659116620166481667716706167341676316792
16820168491687816906169351696416992
17021
17050
17078
1710717136
17164
7193
7222
7250
1172 791730811 336. 17365
. 98769
. 98764. 98760
9874698741987379873298728
98723
9871898714
98709
98704
98700
9869598690986869868 198676
98671986679866298657986529864898643
9863898633
98629
98624986199861498609986049860098595
98590
98585
98580
98575
98570
98565985619855698551
9854698541
9853698531
9852698521
98516985119850698 501
. 98496
. 98491
. 98486
. 9848 1
Cos in S ine CHNa
OO
fi
CJI
GQ
CD
CD
0
1
23
45
6789
10
11
40'
59
89
3
9915
93
93
859
99
992
19
899
8
1 0 °
S ine zCosin Sine Cosin S ine “
Cosin. 98 163. 98 157. 98 152
. 98 146
. 98 140
. 98 135
. 98 129
. 98 124
. 981 18
. 98 112
. 98107
. 98 101
. 98096
. 98090
. 98084
. 98079
. 98073
. 98067
. 98061
. 98056
. 98050
. 98044
. 98039
. 98033
. 98027
. 98021
. 98016
. 98010
. 98004
. 97998
. 97992
. 97987
. 9798 1
7365
. 17393 98476
. 17422
18395
18509
1853818567
185
95
18624
18652
1868 1
18710
18738
18767
18795
1882418852
1888 1
18910. 18938. 18967
. 9848 1
98471
98218
98212
98207
98201
98 196
. 98 190
79°
TABLE X — SINES AND COSINES.
. 1908 1
. 19109
. 19138
. 19167
. 19195
. 19224
. 19252
. 1928 1
. 19309
. 19338
. 19366
. 19395
. 19423
. 19452
. 1948 1
. 19509
. 19538
. 19566
. 19595
. 19623
. 19652
. 19680
. 19709
. 19737
. 19766
. 19794
. 19823
. 19851
. 19880
. 19908
. 19937
. 19965
. 19994
.20022
.20051
. 20079
. 20108
. 20136
. 20165
.20193
.20222
. 20250
. 20279
. 20307
. 20336
. 20364
97975
9796997963
. 97952
. 97946
97940
97934
97928
97922
97916
97910
97905
9789997893
97887
9788 197 75
9786997863
. 97857
78 °
. 20848
. 20877
0 . 97496l
124164.24 192
Cos in. 97437
. 97430
. 9718 9
. 9 7 182
. 97162
. 1 155
. 97148
. 97127
. 97120
7113:971069
7
100
97093
072
l. 96756
'
. 25713
. 25741
. 25769
.257
96793967869677896771
|
OHNJ
OO
Q
OI
O>
9 9 » Q
gHHHHH—lMM
H
10
20
10
CO
MM
Sine Cosin. 42262
. 42288
. 42315
.42341
. 42367
.42 394
. 42420
. 42446
. 42473
. 42 199
.42525
. 42552
. 42578
. 42604
.42631
.42657
. 42683
. 42709
. 42 736
. 42762
. 42788
. 428 15
.42841
. 42867
.42894
. 42920
. 42946
. 42972
. 42 999
. 43025
. 43051
. 43077
. 43 104
. 43 130
. 43156
.43182
.43209
. 43235
.43261
. 43287
.43313
.43340
. 4336643 392
. 43418
43445. 43471
. 43497
. 43523
. 43549
. 43575
. 43602
. 43628
. 43654
. 43680
. 437063733
.43759
.43785
.438 11
. 43837
. 90631
. 90618
. 90606
. 90594
. 90582
. 90569
. 90557
. 90545
. 90532
, 90520
. 90507
. 90495
. 90483
. 90470
. 90458
. 90446
. 90433
. 90421
. 90408
. 90396
. 90383
.90371
. 90358
90346
. 90334
. 9032190309
. 90296
. 90284
. 9 1271
. 90259
. 90246
. 90233
. 90221
. 90208
. 90196
. 90183
. 9017 1
. 90158
. 90146
.90133
. 90120
. 90108
. 90095
. 90082
. 90070
. 90057
. 90045
. 90032
. 90019
. 90007
. 89994
. 8998 1
. 89968
. 89956
. 89943
. 89930
. 89918
. 89905
. 89892
. 89879
Cos in Sine Cos in
TABLE X K‘ SINES AND COSINES.
Sine Cosin Sine Cosin.43837
.4386343889
.43916
.43942
. 43968
.43994
. 44020
.44046
.44072
.44098
.44124
.44 151
.44177
.44203
. 44229
. 44255
. 44281
.44307
. 44333
.44359
.44385
. 44411
. 44437
. 44464
. 44490
.44516
. 44542
. 44568
. 44594
. 44620
.44646
.44672
.44698
. 44724
. 44750
. 44776
. 44802
. 44828
. 44854
. 44880
.44906
. 44932
.44958
. 44984
.45010
.45036
.45062
.45088
. 45114
. 45140
.45166
.45192
. 45218
.45243
.45269
. 45295
. 45321
.4534745373f45399
. 89879
. 89867
. 89854
. 89841
. 89828
. 89816
. 89803
. 89790
. 89777
. 89764
. 89752
. 89739
. 89726
. 89713
. 89700
. 89687
. 89674
. 89662
. 89649
. 89636
. 89623
. 89610
. 89597
. 89584
. 89571
. 89558
. 89545
. 89532
. 89519
. 89506
. 89493
. 89480
. 89467
. 89454
. 8 9441
. 89428
. 89415
. 89402
. 89389
. 89376
. 89363
. 89350
. 89337
. 89324
. 89311
. 89298
. 8 9285
. 89272
. 89259
. 8 9245
. 89232
. 89219. 89206. 89193
. 8 9180
. 8 9167
. 8 9153
. 8 9140
. 8912 7
. 89114
. 89 101
.45399
. 45425
. 45451
. 45477
. 45503
. 45529
. 45554
. 45580
. 45606
. 45632
.45658
.456@
. 45710
. 45736
. 45762
.45787
. 45813
.45839
.45865
. 45891
. 45917
.45942
.45968
. 45994
. 46020
. 46046
.46072
. 46097
. 46123
. 46149
. 46175
.46201
. 46226
. 46252
.4627
. 46304
.46330
.46355
. 46381
. 46407
.46433
.46458
.46484
. 46510
. 46536
. 46561
. 46587
.46613
. 46639
.46664
. 46690
. 46716
.46742
.46767
.46793
.46819
.46844
.46870
.46896
.46921
.46947
Sine Cos in ~
Sine
. 89101
.89087
. 89074
. 89061
. 89048
. 89035
. 89021
. 89008
. 88995
. 8898 1
. 88968
. 88955
. 88942
. 88928
. 88915
. 88902
. 88888
. 88875
. 88862
. 88848
. 88835
. 88822
. 88808
. 88795
. 88782
. 88768
. 88 755
. 88741
. 88728
. 88715
. 88 701
. 88688
. 88674
. 88661
. 88647
. 88634
. 88620
. 88607
. 88593
. 88580
. 88566
. 88553
. 88539
. 88526
. 88512
. 88499
. 88485
. 88472
. 88458
. 88445
.88431
.88417
.88404
. 88390
.88 7 77
. 88363
. 88349
. 88336
.88322
. 88308
. 88295
2 8 °
Sine Cos in Sine Cosin. 46947
. 46973
.46999
. 47024
.47050
.47076
.47101
.47127
.47153
.47178
.47204
.47229
.47255
.4728 1
.47306
.47332
.477 7
8
.47383
.47409
.47434
.47460
. 47486
.47511
.47537
. 47562
.47588
.4 7614
. 47639
. 47665
.47690
.47716
.47
1 741
W677793
24731847844
.47869
. 47895
7920
.47946
.47971
. 47997
. 48022
. 48048
. 48073
. 48099
. 48 124
. 48 150
. 48 175
. 48201
. 48226
.48252
. 48277
. 48303
.48328
. 48354
.4837948405
.48430
. 48456
. 4848 1
. 88295
. 8828 1
. 88267
. 88254
. 88240
. 88226
. 88213
. 88199
. 88185
. 88 172
. 88158
. 88144
. 88 130
. 88117
. 88103
.88089
. 88075
. 88062
. 88048
. 88034
. 88020
. 88006
. 87993
. 87979
. 87965
. 87951
. 87937
. 87923
. 87909
.87896
. 87882
. 87868
. 87854
. 87840
. 87826
. 87812
. 87798
W841~ f~w1 “
. 87756
.SW
. 877297715
8 77018 76878 7 7387659
. 87631
. 87617
. 87603
.875891 0 1 0
75617546
. 87532
. 875187504
. 87490
. 87476
. 87462
Cos in Sine
.4348 1
. 48506
. 48532
. 48557
. 48583
. 48608
. 48634
.48659
.48684
. 48710
.48735
. 48761
. 48786
. 48811
. 48837
. 48862
. 48888
. 48913
. 48938
.48964
. 48989
.49014
. 49040
. 49065
. 49090
. 49116
.49141
.49166
.49192
. 49217
. 49242
.49268
.49293
.49318
.49344
. 49369
. 49394
. 49419
.49445
. 49470
.49495
.49521
.49546
. 49571
.49596
.49622
.49647
.49617
2
.49697
. 49723
.49748
.49773
. 49798
.49824
. 49849
. 49874
.49899
. 49924
. 49950
. 49975
.50000
. 87462
.87448
. 87434
. 87420
. 87406
.87391
. 87377
. 87363
. 87349
. 87335
.87321
. 87306
. 87292
. 8727
. 87264
.87250
.87235
. 87221
. 87207
. 87193
.87178
.87164
. 87150
. 87136
. 871217107
.87093
. 87079
. 87064
. 87050
. 87036
. 87021
.8 7007
. 86993
. 86978
. 86964
. 86949
. 86935
. 86921
. 86906
. 86892
. 86878
. 86863
. 86849
. 86834
. 86820
. 86805
. 86791
. 86777
. 86762
. 86748
. 86733
. 86719
.86704
. 86690
. 86675
. 86661
.86646
. 86632
.86617
. 86603
Cosin S ine
Gb
‘Q
CO
O
O
(N
OV
10
63
HHHHv—LHb—H—lN
MMMN
MMIO
MM
00
00
03
00
00
03
03
00
00
A
A
1k
Q
Q
Q
Q
Q
07
0?
TABLE X .
— SINES AND COSINES.
30° I 3 1°
Sine Cos in S ine
(0
10
N)
MMMM
?O
HHHHHHHHs-s
H
gwmfia
g
a
wwv-o
o
co
oo
q
ca
owfi-mm
v-soco
oo
qcs
onnwwuo
'
CO
g
g
fi
fi
“Q
OV
A
vA
flk
dk
yh
u-A
A
A
Q
O
fic
m’Q
Q
Cfl
fi
O‘D
MH
.50000
.50025
.50050
.50076
.50101
.50151
.50176
.50201
.50227
.50252
.50277
.50302
.50327
.50352
.50377
.50403
.50428
.50453
.50478
.50503
.50528
.50553
.50578
.50603
.50628
.50654
.50679
.50704
.50729
.50754
.5077
.50804
.50829
.50854
.50879
.50904
.50929
.50954
.50979
.51004
.51029
.51054
.51079
.51104
.51129
.51154
.51179
.51204
.51229
.51254
.51279
.51304
.51329
.51354
.51379
.51404
.51429
.51454
.51479
.51504
. 86603
. 86588
. 86573
. 86559
. 86544
. 86530
. 86515
. 86501
. 86486
. 86471
. 86457
. 86442
. 86427
. 86413
. 86398
. 86384
. 86369
. 86354
. 86340
. 86325
. 86310
. 86295
. 8628 1
. 86266
. 86251
. 86237
. 86222
. 86207
. 86192
. 86178
. 86163
. 86148
. 86133
. 86119
. 86104
. 86089
. 86074
. 86059
. 86045
. 8603086015
. 86000
. 85985
. 85970
. 85956
. 85941
. 85926
. 85911
. 85896
. 8588 1
. 85866
. 85851
. 85836
. 85821
. 85806
. 85792
. 85777
. 85762
. 85747
. 85732
85717
Cos in S ine
.51504
.51529
.51554
.5157
.51604
.51628
.51653
.51678
.51703
.51728
.51753
.51778
.51803
.51828
.51852
.51877
.51902
.51927
.51952
.51977
.52002
.52026
.52051
.52076
.52101
.52126
.52151
.52175
.52200
.52225
.52250
.52275
.52299
.52324
.52349
.52374
.52399
.52423
.52448
.52473
.52498
.52522
.52547
.5257
.52597
.52621
.52646
.52671
.52696
.52720
.52745
.52770
.52794
.528 19
.52844
.52869
.52893
.52918
.52943
.5296752992
Cos in
tu-k
“kw
co
m
m
00
00
0)
C0
10
10
10
10
10
10
10
10
MHHH
HHh—l
fi
wwr-to
wg-Qm
ot
sf
wwa
0
9
576
-405
01
»;
moco
oo
sz
ca
oms
s‘ o
z
:Sco
oo
-at
koo
wuo
l
Aor
UY
OY
OY
U!
U
fi
g
flk
fi
CD
CD
“QC?
UV
OI
£886
3
3
TABLE X .
— SINES A-ND COSINES.
S ine Cosin S ine Cosin.64279.64301.64
3
23.64346.64368.64390.64412.64435.64457.64479.64501
.64524.64546.64568.64590.64612.64635.64657.64679.64701.64723
.64746
.64768
.64790
.648 12
.64834
.64856
.64878
.64901
.64923
.64945
.64967
.64989
.65011
.65033
.65055
.65077
.65100
.65122
.65144
.65166
.65188
.65210
.65232
.65254
.65276
.65298
.65320
.65342
.65364
.65386
.65408
.65430
.65452
.65474
.65496
.65518
.65540
.65562
.65584
.65606
. 76604
. 76586
. 76567
.76548
. 76530
. 7651 1
. 76492
.76473
. 76455
. 76436
. 76417
. 76398
. 76380
. 7636176342
. 76323
. 76304
. 76286
. 76267
. 76248
. 76229
. 76210
. 76192
. 76173176154. 76135. 761 16. 76097. 76078. 76059
. 76041
. 76022
. 76003
. 75984
. 75965
. 75946
. 75927
. 75908
. 75889
. 75870
. 75851
; 75832. 758 13
. 75794
. 75775
. 75756
. 75738
. 75719
. 75700
. 75680
. 75661
. 75642
. 75623
. 75604
. 75585
. 75566
. 75547
. 75528
.75509
. 75490
. 75471
C—
os in S ine
.65606 .75471
65628 .75452
.65650 . 75433
.65672
.65694
.65716
.65738
.65759
.65781
.65803
.65825
.65847
.65860
.65891
.65913
.65935
.65956
.65978
.66000
.66022
.66044
.66066
.66088
.66109
.66131
.66153
.66175
.66197
.662 18
.66240
.66262
.66284
.66306
.66327
.66349
.66371
.66393
.66414
.66436
.66458
.66480
.66501
.66523
.66545
.66566
.66588
.66610
.66632
.66653
.66675
.66697
.66718
.66740
.66762
.66783
.66805
.75414
. 75395
. 75375
. 75356
. 75337
. 75318
. 75299
. 75280
. 75261
. 75241
. 75222
. 75203
. 75184
. 75165
. 75146
. 75126
. 75107
. 75088
. 75069
. 75050
. 75030
. 75011
. 74992
. 74973
. 74953
, 74934
. 74915
. 74896
. 74876
. 74857
. 74838
. 748 18
. 74799
. 74780
. 74760
. 74741
. 74722
. 74703
. 74683
. 74664
. 74644
. 74625
. 74606
.74586
. 74567
. 74548
. 74528
. 74509
. 7448974470
74451
74431744 12
.66891 . 74334
166913 . 74314
Cofl n Sine
S ine.669 13
.66935
.66956
.66978
.66999
.67021
.67043
.67064
.67086
.67107
.67129
.67151
.67172
.67194
.67215
.67237
.67258
.67280
.67301
.67323
.67344
.67366
.67387
.67409
.67430
.67452
.67473
.67495
.67516
.67538
.67559
.67580
.67602
.67623
.67645
.67666
.67688
.67709
.67730
.67752
.67773
.67795
.67816
.67837
.67859
.67880
.67901
.67923
.67944
.67965
.67987
.68008
.68029
.68051
.68072
.68093
.68 115
.68 136
.68 157
.68 179
.68200
Cosin
X I.
-NATURAL SECA’NT S AND COSECANTS.
0 0
00000
00000
00000
00000
00000
. 00001
00001
00001
0000100001
'
00001000010000200002
. 00002
00002
00005
00005
00008
00009
00010
00010
000 12
00012
00014
00014
00015
SECANTS
1 0 2 0 3 0
00016 00062 00 139
00016 00063 00 140
000 17 00064 00142
00065 00143
00018 00066“a 00 145
000 18 00067 00 147
00019 00068 00148
00020 00 150
00020 00070 00151
00021 00072 00153
.00021 .00155
00022 00074 00156
00023 00075 0015800023 00076 00159
00024 00077 00161
00024 00078 00163
00025 00079 00164
00026 0008 1 00166
00026 0016800027 00083 00169
. 00028 . 0017100028 00 17300029 00087 00175
00030 00088 00176
00089 00 17800031 00090 00 180
00032 00091 00182
00033'
00183
00094 00 185
00187
.00035 . 00 189
00036 00190
0003 00 192
00037 00 194
00102 00 196
00103 00198
00041 00106 0020100203
00108 00205
. 0004300044 001 1 1 0020900045 00 113 0021 1
00046 001 14 0021300047 001 15 00215
00117 1 0021600118 . 00218
00120
00050 00 12100051 00122
00053 00125‘
00127
00128
00 130
00057 00131 0023600133 . 00238
00059 00 134 00240
00060 00136 0024200061 00137 00244
COSECANT S .
4° 5°
00246 00385
00392
0039500397
0026100405
00265 00408
. 0026700413
00271 004 1600274
00276 00421
.0027800280 0042700282 0042900284 00432
00287
.00438
00291 00440
00443
00296 0044600298 0044900300 00451
00302 00454
00305 0045700307 0046000309 00463
. 00312
00314
00316 00471
00318 00474
0047700480
00326 0048200328 . 00485
00333
.00335
00337 0049700340
00503
0050600347 00509
00512
00352 0051500518
00521
0052700364 00530
0036700369 0053600372 0053900374 0054200377 0054500379 0054800382 00551
60
00554
0055700560
005690057300576005700582
005920059500598
00604
.00617
0062400627
0063400637006400064400647
.0065000654
00660
00667
0067400677
0069 100695006980070100705
00708
00712
00715
00722
00726
00730
00737
00740
00747‘
510
. 07 1157 12607 13807 150
07174
07 1867199
072 11
07223
7235
. 07247
07259
0727 1
07283
07307
07320
07332
07344
07356
.0736807380
07393
07405
07417
07429
07442
07454
07466
07479
. 0749107503
07516
7528
07540
07553
07565
075707590
07602
.076150762707640076520766507677076900770207715077
.077400775207765077780779007803
078 1607828
0784107853
COSECANTS
X I . -NA'
1‘
URAL SECANTS AND COSECANTS .
SECANT S .
23 ° 24 ° 25° 26° 2 7°
09478 10353 1 127 12249
08663 09492 10368 11292 1226608676 10383 1 1308 12283
08690 09520 10398 1 1323 12299
08703 09535 10413 1 1339 1231608717 10428 1 1355 12333
08730 09563 10443 1 1371
08744 0957 10458 1 1387 1236608757 09592 10473 1 140308771 10488 1 14 19
. 08784 .09620 . 10503 . 1 1435 . 1241608798 10518 11451 12433
088 11 09649 10533 1 146708825 09663 10549 1 1483 12467
09678 10564 1 149908852 09692 10579 1 1515
08866 09707 10594 1 1531 1251809721 10609 1 1547 12534
088 93 09735 10625 1 1563 12551
08907 09750 10640 1 1579 12568
. 08921 . 09764 . 10655 . 1 1595 . 12585
08934 0977 10670 1 161 1 1260208948 09793 10686 1 1627 12619
08962 09808,10701 1 1643 12636
08975 09822 10716 1 1659 1265308989 09837 10731 1 1675 12670
09851 10747 1 1691 12687
09866 10762 1 1708 12704
09880 10777 1 1724 12721
09895 10793 11740 12738
. 09058 . 09909 . 10808 . 1 1756 . 127
55
09072 09924 1 1772 1277
09086 09939 10839 1 1789 12789
09953 10854 1 1805 12807
09968 10870 1 1821 12824
09127 10885 11838
0914 1 09997 10901
09 155 100 12 109 16 1 1870 12875
09169 10932 1 1886 12892
09183 10947 1 1903 129 10
. 09197 . 10055 . 10963 . 1 1919
09211 10071 10978 11936
10994 1 1952 12961
09238 10100 1 1009 1 1968 12979
09252~
10115 1 1985 12996
10 130 1 1041 12001 13013
10144 11056 12018 1303 1
09294 10159 1 1072
09308 10174 1 1087 12051 13065
09323 10189 11103 13083
. 09337 . 10204 . 111 19 . 12084 . 13100
10218 1 1134 12 100 13 117
09365 1 1150 121 17 13135
09379 10248 1 1166 12133 13 152
09393 10263 1118 1 12150 13170
09407 11197 12166 13187
09421 10293 1 1213 12183 13205
09435 10308 1 1229 12 199 13222
09449 10323 1 1244 12216
09464 10338 1 1260 12233 13257
66° 65° 64 ° 63 ° 62°
418 °
13275
13292
133 10
133 37
13345
13362
13398
134 15
13433
. 13451134681348613504
'
13521
13539
13557
13593
13610
. 136281364613664
1368213700
137 1813735
13753
1377 1
13 789
. 1380713825
13843138611387913897139 1613934
13952
13970
. 13988
1402414042140611407914097141 15
14 134
14 152
. 14170
1418814207
1 4225
14243
14262
14280
14299
14317
14335
61 °
119 °
14354
14372
1439 1
14409
14 42814446
1 1465
14483
1 1502
1 152 1
. 14539
14558
14576
14595
14614
14632
14651
14670
1 4689
14707
. 14726
147451476414782148011482014839
14858
1487714896
14933
14952
1497 1149901500915028
1506615085
. 15105
1512415143
151621518 115200
152191523915258
15277
. 1529615315
15335
15354
15373
15393
15431
15470
60 °
COSECANTS .
SECANTS .
30 ° 3 1 °
15189 1668 4
15509 16704
15528
15548 16745
1676616766
16806
15626 16827
15645‘
16848
15665 16868
. 15684 . 16889
1&w915724 1693015743 1695015763 1697 1
15782 16992
15802 170 12
15822 17033
15841 17054
15861 17075
. 1588 1 . 17095
15901 17116
15920 17 137
15940 17158
15960 17 178
15980 1 7 199
16000 17220
1601916039 17262
16059 17283
. 16079 7304
16099 17325
16119 1734616139 1736716159 1738816179 17409
16199 17430
16219 17451
16239 17472
16259 17493
. 16279 17514
16299 17535
16319 1755616339 757716359 759816380 1762016400 1764 116420 1766216440 1768316460 17704
. 1648 1 177261650 1 177 716521 1776816541 7790
16562 1781 116582 1783216602 1785416623 17 75
16643 17896
16663 179 18
59 ° 58 °
32 °
17939
1796117982
18004
18025
18047
18068
18090
18 11 1
18 133
. 18 155
18 17618 19818220
1826318285
1830718328
18350
. 1837218394184 16
184371845918 611850318525
1854718569
. 18591186131863518657186791870118723187451876718790
. 188121883418856
188781890 118923189451896718990
19012
. 1903419057
190791910219124191461916919 19 119214
19236
57°
233 °
19259
1928 1
19304
1934919372
1939 4
194 17
19440
19463
. 19485
19508
19531
19554
1957619599
19622
1966819691
. 197 13197361975919782
1980519828
19851
19874
1989719920
.19944
1996719990
20013
20059
2012920152
. 20176
20222
202462026920292
20316203392036320386
.2041020433
20457
20527
20575
2059820622
56°
XL— NATURAL SECANT S AND COSECANT S. 7
3 4 °
20669
20740
20764
208 12
208 36
20859
.208832090 42093 1
20955
20979
2100321027
210512107521099
. 21 123
2114721 17 12 1 195
21220
21244
2 126821292
21316213 41
.2136521389
214 14
21438214622148721511
21535
215602 1584
.216092163321658216822170721731
217562 178 1
21805
21830
.2 1855
21879
21929
2195321978
22028
22053
22 77
55°
$31 1
o
—dm
wa
ct
ca
q
oo
co
l
XI .
— NATURAL SECANTS AND COSECANT S.
SECANTS .
56° 57° 58 ° 59 ° 60 ° 61 °
78906 83690 88796 94254 0010 1 06375
78984 83773 88884 94349 0648379061 88972 94443 06592
79138 89060 94537 00404 06701
79216 84020 89148 94632 0680979293 84103 89237 94726 00607 06918
79371 84 186 89325 94821 0 7 027
79449 84269 894 14 94916 008 10 07137
79527 84352 89503 9501 1 009 12 0724679604 84435 89591 95106 01014 07356
. 79682 . 89680 . 95201 .
79761 84601 89769 95296 01218 07575
79839 84685 89858 95392,
01320 0768579917 84768 89948 95487 0 1422 07795
79995 84852 90037 95583 01525 07905
80074 84935 90126 95678 01628 08015
80152 85019 90216 95774 01730 08 12680231 85103 90305 95870 01833 0823680309 85187 90395 95966 01936 08347
90485 96062 02039 08458
. 80467 . 90575 . 96158 2 0856980546 85439 90665 96255 02246 0868080625 85523 90755 96351 02349 0879180704 85608 96448 02453 0890380783 85692 90935 96544 02557 0901480862 85777 91020 96641 02661 09126
80942 85861 9 1116 96738 027
65 09238
8 1021 85946 9 1207 96835 02869
8 1101 86031 91297 96932 02973 0946281 180 86116 91388 97029 03077 09574
. 81260 . 91479 . 97127
8 1340 86286 9 1570 97224 03286 09799
8 1419 86371 91661 97322 03391 09911
8 1499 86457 91752 97420 03496 10024
8 1579 86542 91844 97517 03601 10137
8 1659 86627 91935 97615 . 03706 10250
8 1740 86713 92027 97713 038 1 1 103638 1820 86799 92118 978 11 03916 104778 1900 86885 92210 979 10 04022 10590
8 198 1 86990 92302 98008 04128 10704
. 82061 . 92394 . 98107 . 108 1782142 7142 92486 98205 04339 10931
82222 7229 9257 98304 04445 1 1015
8 2303 87315 92670 98403 04551 1 1159
8238 4 87401 92762 98502 04658 1 1274
82465 87488 92855 9860 1 04764 1 1388
82546 8757 92947 98700 04870 1 1503
82627 8 7661 93040 98 7 99 04977 1 161782709 87748 93133 98899 05084 1 173282790 87834 93226 98998 05191 11847
. 82871 . 93319 .99098 . 1196382953 88008 93412 99198 05405 1207883034 88095 93505 99298 05512 12193
83116 88 183 93598 99398 05619 12309
83198 8 8270 93692 99498 05727 1242583280 88357 93785 99598 05835 1254083362 88445 938 79 99698 05942 1265783444 88532 93973 99799 06050 1277.
83526 88620 91066 99899 06158 128898 3608 88708 94 160 06267 13005
3 3° 32 ° 3 1 ° 30 ° 2 9 ° 2 8 °
COSECANTS .
10
62 °
2 1300513 122
13239
1335613473
13590
13707
13825
13942
14178
. 1429614414
14533
1465114770
14889
1500815127
1524615366
. 15485
15605
15725
15845
15965
16085
16206163261644716568
. 16689168 10169321705317175
1729717419
1754 1
1766317786
17909
18031
18 154182771840 1
18524
1864818772
18895
19019
. 19144
1926819393
19517
19642
197671 9892
2001820143
20269
2 7°
31 8 XI .
— NATURAL SECANTS AND COSECANT S .
XL— NAT URAL SECANTS AND COSECANT S.
8 4 °
59332
62002
6468767387
70103
72833
75579
78341
8 1119
839 12
89547
92389
95248
98 123
03923
09792
12752
18725
21739
2477
278 19
30887
33973
37077
4020 1
43343
4968552886561065934662605658856918672507
75849
825968600189428
92877
99841
0689410455
176462 127724932
2861032313
4356947371
50
85°
51 199
55052
58932
6283766769
70728
74714
7 727
827
6886837
99214
076101 1852
16125204272476129125
3794842408
46900
51424
55982
605765197
6985674550
84042
8884 1
9367798549
08040
13388
1841 123472
33712
38891441 12
49373546766002 1654087083876312
8739192999
98651
10096
217302762033559
4 0
SECANTS .
86° 8 7°
19 10732
39547 21397
45586 32182
51676 43088
578 17
640 11 65275
70258 76560
76558 87976
829 13 99524
89323
95788 23028
08890 47093
15527 59341
22223 7 1737
28979 84283
35795 96982
42672 21 0983849611 22852
56614 3602763679 49368
78005 7655585268
92597
99995 1865333050
14999 4763522607 624 1330287 7738638041 92559
5377'
23520
61751 3931669808 5532977944 7156386159 88022
9445621637
1 1297 3880219843 56212
37 196 9 1790
4600554903 28414
63893 47134
72975 6613282152 85417
91424
2486910262 45051
29501 8636039274
49 153 2898 1
59 139 5080472978
79438 95513
8 7554 1700
10732 65371
3 ° 2 °
COSEC ANTS
8 8 °
89440
388 12
6413789903
42802
699607607
83623
4367 174554
38 118
70835
38232
44539
81452
57633
96953
78185
51855
97797
92963
9277‘
9757 1
64980
8 4026
74997
89 156
1 0
8 9lo
319
320 TABLE K IL — T ANGE'
fi Ts AND COTANGENTS.1 0
Cotang Tang;
42 9641
42 4335
40 435839 965539 505939 056838 617738 1885
36 9560365627
32 49 1332 118 131 82053 1 528431 2416
30 959930 683330 411630 144629 8823
29 624529 3711
29 1220
Tang
.03492
.03521
.03550
.03579
.03609
.03638
. 03667
.03696
. 03725
.03354
.038 12
. 03842
.03871
. 03900
.03929
. 03958
. 03987
.04016
. 04046
.04075
.04104
.04133
. 04162
. 04191
. 04220
. 04250
. 04279
.04308
.04337
. 04366
. 04395
.04424
. 04454
.04483
. 04512
.04541
.04570
.04599
. 04628
.04658
.04687
.04716
.04745
.04774
.04803
.04833
.04862
. 04891
.04920
.04949
.04978
.05007
.05037
.05066
.05124
.05153
. 05182
.05212
.05241
Cotang
27 489927 2 71527 056626 8450266367
26 030725 834825 6418
24
24
24
24 195724 0263
23 371823 213723 057722 9038
22 751922 6020
£5253
Tang0 . 14054
1 . 14084
2 . 1 1113
3 . 14143
4 . 14173
5 . 14202
6 . 14232
7 . 14262
8 . 14291
9 . 143210 . 14351
1 1 . 1438 112 . 144 10
13 . 14440
14 . 14470
15 . 14499
16 . 14529
17 . 14559
18 . 14588
19 . 1461820 . 14648
21 . 1467822 . 14707
23 . 14737
24 . 1476725 . 1479626 . 1482627 . 1485628 . 1488629 . 1 4915
30 . 14945
31 . 1497532 . 15005
33 . 1503434 . 1506435 . 1509436 . 15124
37 . 15153
38 . 1518339 . 15213
40 . 15243
41 . 1527242 . 15302
43 . 15332
44 . 1536245 . 15391
46 . 15421
47 . 15451
48 . 1548 149 . 15511
50 . 15540
51 . 1557052 . 1560053 . 1563054 . 1566055 . 1568956 . 157197 . 1574953 . 1577933 . 15809LO . 15838
TABL E XII .
‘
ANUENTS AND COTANGENTS.
7 115377 1m %7 085467 070597 055797 041057 026376 911746 997 186 98268696823
6 953856 939526 925256 911046 896886 882786 868746 854756 840826 82694
6 8 13126 799366 785646 771996 758386 74483
75
Gotang Tang_
15838
15868158981592815958
159881601716047160771610716137
16167. 16196. 16226. 16256. 16286. 16316. 16346. 16376. 16405. 16435
. 16465
. 16495
. 16525
. 16555
. 16585
. 16615
. 16645
. 16674
. 16704
. 16734
. 16764
. 16794
. 16824
. 16854
. 16884
. 16914
. 16944
. 16974
. 17004
. 17033
. 170637093
. 17123
. 17153
. 171837213
. 17243
. 17273
. 17303
. 17333
. 17363
. 17393
. 17423
. 17453
. 17483
. 17513
. 17543
. 17573
. 1760317633
Gotang Tang Gotefi Tang
80°
’
10 °
g ang Cotang Tang Cotang
. 17633 . 19438
. 17663 . 19468
. 17693 . 19498
. 17723 . 19529
. 17753 . 19559
. 17783 . 1958 97813 . 1961917 19649 5 08921
. 17873 . 19680
. 17903 . 19710
. 17933 . 19740
. 19770
. 198015.54851 198315.53927 19861 5 03499
. 19891
. 19921
. 19952
19 982 5 0045120012
. 20042
.20073
. 20103
.201335 44857 20164
.20194
.20224
. 20254
. 20285. 20315
5 39552 .20345
5 38677 .203765 37805 .204065 36936 .20436
. 20466
.204975 34345 .205275 33487 .20557
‘
5 32631 .205885 3177 .206185 30928 .20648
5 30080 .206795 29235 .207095 28393 . 207395 27553 . 207705 26715 . 208005 25880 . 208305 25048 . 208615 24218 .208915 23391 . 209215 .20952
5 .209825 .210135 .210435 .210735 .211045 .211345 16863 .211645 16058 .21195
. 21225
.21256 4 . 70463
79°
Tang Cotang Tang7?
8
8
3
33888!
[
OHMCO
A
O'
Q'Q
CDQO
TABLE X11 .
-TANGENTS AN! ) 323
Tang Cotang Tang Gotang Tang_
Cotang Tang Co tang
0 .21256 4 . 70463 4 . 33148 .24933 4 . 01078 .26795 60
1 .21286 23117 . 24964 3 . 72771 59
2 .21316 .23148 . 26857 58
3 21347 23179 4 31430 3 . 99592 57
4 21377 . 23209 . 25056 565 21408 . 25087 26951 55
6 21438 .23271 25118 26982 54
7 21469 25149 27013 538 21499 25180 7044 52
9 21529 . 25211 51
10 21560 50
11 21590 .27138 49
12 .21621 . 25304 . 27169 48
13 .21651 .25335 . 27
201 47
14 .21682 .23516 . 25366 .27232 46
15 .21712 .23547 .25397 .27263 45
16 23578 3 93271 3 .66376 44
17 .21773 . 23608 .25459 43
18 .21804 . 23639 . 25490 42
19 .21834 4 .58001 .23670 25521 3 . 91839 27388 41
20 .21864 . 23700 . 25552 40
21 .21895 .23731 . 25583 3922 21925 23762 4 . 20842 25614 3 90417 27482 3 638 74 3823 . 21956 . 23793 . 25645 27513 3724 .21986 . 25676 27545 3625 .22017 .23854 .25707 27576 3526 . 22047 . 23885 .25738 27607 3427 .23916 . 25769 7638 3328 . 22108 . 23946 . 25800 27670 3229 22139 4 51693 23977 4 17064 3 87136 27701 3 60996 3130 .22169 .25862 .27732 3 60588 30
31 .24039 .277 2932 .22231 . 24069 .25924 .27795 2833 22261 . 24100 4 14934 25955 3 . 85284 3 59370 2734 . 24 131 .25986 2635 .22322 .24162 .27889 2536 . 24193 . 26048 .27921 M37 26079 .27952 2338 .22414 4 .46155 .24254 . 26110 .27983 2239 . 24285 . 26141 .28015 2140 .26172 2
4 1 .24347 . 26203 1942 .22536 .24377 26235 3 .55761 1843 24408 26266 28 140 1744 .22597 . 24439 .26297 1645 .22628 4 . 4 1936 .244 70 4 .08666 3 . 79827 1546 . 22658 4 41340 .24501 6359 3 79378 14
47 22689 .24532 6390 3 . 78931 28266 13
48 .22719 4 40152 .24562 4 07127 6421 28297 12
4 39560 24593 4 06616 3 . 78040 28329 11
. 24624 .26483 .28360 10
.26515 3 . 7 7 152 . 28391 3 .52219 926546 8
24717 2657 3 .51441 74 . 36623 4 04081 26608 3 .51053 64 36040 2477 4 .03578 26639 28517 3 .50666 54 35459 26670 3 . 74950 4
22995 4 34879 26701 3 74512 3
.23026 .24871 .28612 2
.23056 4 . 33723 .26764 3 . 73640 .28643 3 . 49125 1
.23087 4 . 33 148 3 . 73205 .28675 3 48741
Gotang Tang Cocang Tang Conan
77° 1 76°
Tang Cotang Tang74°
324 TABLE XII .— TANGENTS AND COTANGENTS.
Tang
. 33266
. 33298
. 33330
. 33363
. 33395
. 33427
. 33460
. 33492
. 33524
. 33557
. 33589
. 33621
. 33654‘
. 33686
. 33718
. 33751
. 33783
. 338 16
. 33848
. 3388 1
. 33913
. 33945
.33978
. 34010
. 3 4043
. 34075
.34108
u34140.34173
.34205
. 34238
. 34270
.34303
.34335
.34368
. 34400
.34433
Ootang
8
8
83
3
33
2
3
32
I
8
882
3
3
2
32
2
TABLE XII .
— TANGENTS AND COTANGENTS.
28 ° . 29 ° 0 30 °
Tang Cotang Tang Co tang Tang Gotang Tang Gotang53171 1 88073 1 80405 57735 1 73205 60086 60
.53208 1 . 7941 55469 57774 1 . 73089 .60126 59
.53246 1 87809 .55507 1 . 80158 57813 1 72973 60165 5853283 1 7677 55545 1 . 80034 57851 1 72857 60205 1 66099 57
.53320 1 7546 55583 1 . 79911 57890 1 . 72741 56
.53358 1 87415 55621 17
9788 57929 1 72625 55
.53395 1 87283 1 79665 57968 1 72509 54
53432 1 87152 55697 1 79542 1 72393 1 65663 53
53470 1 87021 55736 1 . 79419 52
.53507 5577 1 72163 60443 51
.53545 .55812 1 . 79174 .58124 60483 50
53582 1 . 86630 55850 1 . 79051 58 162 60522 1 .65228 49
.53620 1 . 86499 1 78929 58201 1 . 718 17’ 60562 1 .65120 48
53657 1 86369 1 78807 58240 1 71702 60602 1 65011 47
53694 1 86239 55964 1 77
8685 58279 1 71588 60642 4653732 1 86109 1 78563 58318 1 71473 6068 1 1 64795 45
.53769 1 56041 1 78441 1 71358 1 .64687 44
53807 1 85850 56079 1 78319 58396 1 71244 60761 1 .64579 43
.53844 1 85720 56117 1 78198 .58435 1 . 71129 60801 42
53882 1 85591 1 78077 58474 1 71015 .60841 1 .64363 41
.53920 .56194 58513 .60881 1 64256 40
53957 .56232 58552 1 . 70787 60921 39
53995 1 85204 56270 1 . 77713 58591 1 70673 60960 38
54032 56309 1 . 77592 58631 1 70560 61000 37
.54070 1 84946 56347 58670 .61040 36
.54107 56385 1 . 77351 .58709 61080 35
.54145 1 . 84689 56424 1 .77230 58748 1 . 70219 61120 1 63612 34
.54 183 56462 61160 33
54220 1 . 84433 6501 l 76990 1 6999 61200 1 63398 32
54258 1 84305 6539 1 76869 58865 1 6987 61240 1 63292 31
.54296 -1 84 177 6577 58905 61280 1 63185 30
54333 1 . 84049 .56616 1 . 76629 58944 1 69653 61320 1 .63079 29
z5437 1 83922 58983 1 6954 1 61360 28
.54409 56693 1 . 76390 59022 1 .69428 61400 1 .62866 27
.54446 1 83667 .56731 1 76271 59061 1 .69316 61440 1 .62760 26
.54484 1 83540 56769 1 76151 59101 1 69203 61480 1 .62654 25
54522 1 . 83413 56808 1 76032 59140 1 69091 61520 1 62548 24
.54560 1 83286 56846 1 75913 .59179 1 .68979 61561 1 .62442 23
.54597 1 83159 1 75794 59218 1 .68866 601 1.62336 22
54635 56923 1 75675 59258 1 68754 641 1 62230 21
54673 1 82906 1 75556 59297 1 68643 681 l 62125 20
54711 1 . 75437 59336 1 68531 61721 19
54748 1 82654 57039 1 75319 59376 1 68419 61761 1854786 57078 1 75200 .59415 1 .68308 .61801 1 61808 17
54824 7116 1 . 75082 .59454 1 68196 61842 1654862 57155 1 74964 59494 1 68085 61882 1 61598 15
54900 1 82150 57193 1 74846 59533 1 .67974 61922 1 61493 14
54938 1 . 82025 57232 1 . 74728 59573 1 67863 61962 1 61388 13
. 5497 1 . 8 1899 57271 .59612 1 .67752 .62003 12
55013 1 8 1774 59651 62043 1
.55051 1 . 8 1649 .59691 1 67530 62083 0
55089 1 81524 57386 1 74257 59730 1 67419 62124 1 60970 9
55127 1 81399 7425 1 74140 5977 1 7309 62164 8
55165 1 81274 57464 1 74022 59809 1 .67198 .62204 1 .60761 7
55203 1 8 1 150 57503 1 73905 59849 1 67088 62245 1 .60657 6
5524 1 1 8 1025 57541 1 73788 59888 1 66978 1 60553 5
55279 1 80901 57580 1 73671 59928 1 66867 62325 1 60449 4
55317 57619 1 73555 .59967 1 66757 62366 1 60345 3
.55355 1 . 80653 .57657 .62406 2
.55393 1 . 80529 57696 1 73321 1 66538 62446 1 .60137 1
.554'
$1 1 . 8 0405 .57735 I 1 . 73205 7 60086 1 66428 .6248 7 1 .60033 0_
COLang Tang Gotang I Tang Cot ang Tang Got ang TangI
58 °
330
H
HHMHHHP‘
H
76
13
g
eo
gq
ca
m
ux
oo
wfl
oco
oo
-ccws
wwr-to
l
8288
233
12
2
8
8
TABLE X11.— TANGENTS AND COTANGENTS.
40°
T ang | Cotang £21952 . Cotang l .Tang | Go tang
847
85912
8596386014
861158616686216
N
7
86318
MM
HHHHMMHHH
HH 19175
19105
19035
87955
. 88 1 10
. 88 162
. 88214
. 88317
4 1 ° 42°
90040 1 1 106190093 1 1099690146 1 10931
90199 1 1086790251 1 10802
14699 90304 1 1073714632
"
7 1 1067214565 90 41 0 10607
90463 1054390516 10478
9
. 90674
9104691099
9115391206 1
91259 1
91313 191366 1
91419 191473 1
91526 1 09258
91580 1 09195
1 91633 1 09131
1 9168 7 1 7
1 91740 1 090031 917 1 089401 91847 1 088 7
91901 1 088 13
91955 1 08749
1 1 1975
1 1 1909
1 1 1844
1 1 1778
1 1 1713
1 116481 1 15821 1 1517
1 11452
1 1 13871 1 13211 1 12561 1 1 191
1 1 11261 1 1061
Co tang j Tang Go t ang | Tang Cotang Tang
47°49° il 4 8 °
TABLE XII.— TANGENTS A ND COTANGENTS .
0 0n
. 98 327
. 98384. 9844 1. 98499. 98556. 98613
7 1
. 98728
. 98 786
. 98843HHHH
’
HHHH
. 01642
. 01583
. 01524
.01465
. 01406
. 01347
. 01288
. 01229
Gotang Tang
43
39
8
8
852
. 99362
33 1
Ver s ine z Exsec a n‘
r‘z f e c q nf - I
TABLE XIII .
—VERSINES AND EXSECANTS.
TABLE XIII.— VERSINES AND EXSECANTS.
Vers. Exsec . Vet s. Exsec . Vet s. Exsec . Vers. Exsec .
02240
02291
03347
02935
03407
TABLE XIII .— VERSINES AND EXSECANTS.
Vers . Exsec . Vc rs . Exsec . Vers. Exsec . Vers. Exsec .
TABLE XIII .~— VERSINES AND EXSECANTS.
Vers . Exsec . Vers. Exsec . Exsec . Vers . Exsec .
13257
12566
13793
13368
. 15470
. 15489
. 15509
. 15528
. 15548
. 15567
. 15587
. 15606
. 15626
. 15645
15665
156841570415724157431576315782
15802158221584115861
158811590115920
15940
159601598016000019039
059
. 16079
. 16099
. 16119
. 16139
1617916199162191623916259
1627916299
. 16319
. 16339
. 16359
. 16380
. 16400164201644016460
16481165011652116541165621658216602166231664316663
. 14283
. 14298
. 14313
. 14328
. 14343
. 14358
. 14373
. 14388
. 14403
. 14418
.14433
. 14449
. 14464
. 14479
. 14494
. 14509
. 14524
. 14539
. 14554
. 14569
. 14584
. 14599
. 14615
. 14630
. 14645
. 14660
. 14675
. 14706
. 14721
. 14736
. 14751
. 14766
. 14782
. 14797
. 14812
. 14827
. 14843
. 14858
. 14873
. 14904
. 14919
. 14934
. 14949
. 14965
. 14980
. 14995
. 1501 150265041
5057. 15072
. 15087
. 15103
. 15118
. 15134
. 15149
. 15164
. 15180
. 15195
. 16663
. 16684
. 16704
. 16725
. 16745
. 16766
. 16786
q16806
. 16827
. 16848
. 16868
. 16889
. 16909
. 16930
. 16950
. 16971
. 16992
. 17012
. 17033
. 17054
. 17075
. 17116
. 17137
. 17158
. 17178
. 17199
. 17220
. 17241
. 17262
. 17493
. 17535
755675777598
. 1762076417662768317704
1772617747
1776817790178 1117832
178541787517896
17918
TABLE XIII.-VERSINES AND EXSECANTS.
Vers. Exsec . Vers . Exsec . Vers. Exsec.
342 TA BLE XIII .— VERSINES AND EXSECANTS.
Vers.
TABLE XIII.
— VERSINES AND EXSECANTS.
Vers. Exsec . Vers. Exsec . Vers. Exsec . Vers . Exsec .
. 43956. 31821
44304
44391
42293
33087
TABLE XI II.— VERSINES AND EXSECANTS.
Vers . Exsec . Vers. Exsec . Vers .“ _ cfl
TABLE XIII .— VERSINES AND EXSECANTS. 347
Vers. Exsec . Vet s . Exsec . Vet s . Exsec . Vers. Exsec .
1 . 00607.51697
52259
Vers .
T ABLE XIII .- VERSINES AND EXSECANTS.
Exsec . Vers . IExsec . Vers. Exsec . Vers. Exsec .
T ABLE XIII .— VERSINES AND EXSECANTS .
Vers. Exsec . Vet s. Exsec . Vers. Exsec .
96510
. 98371
. 98575
. 9691775 39655
. 96975
20 .62876
97324
99186
99244
39 44820
40 42266
. 9 7615
.960 16 99505
96103
. 99651. 97935
99709
2565546
. 98051. 99825
96510 56 29869
359TABLE XVI. —TRANSIT ION CURVE TABLE.
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G O O C3 0 0 0 O0 1 9 0 v oo m sg $ 8 0 2 8F ‘H N N N CO CO V ‘ V V IO IO
TABLE XVI.— TRANSIT ION CURVE TABLE .360
000
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v-I Vfl
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v-fl fl v—n0 7 0 2 0 2 0 2 0 2 0 2 60 60
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TABLE XVI.
— TRANSIT IOJ CURVE TABLE .362
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v-q v-( v-i v-NN CQ OO CO V 'QH O AO CD ca b oo cm m O H N W V V AO QD E GD
CD G Q w w i ~ 1 ~ © AQ V OO CN O Q I ‘ IQ CQ v-fl G QD V v—w ID H I‘ OO OJ
mm
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w Q O v-‘CQ V IO CDH v-i N OI OO 60 m 1~
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fl v-q v-i fl H v-HN CN N N O‘DO
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v-i v-il v-‘N CQ CQ C’O W V V LO IO co w b oo oo Q O v—fl v-‘ CV co vn o co ts
w b l‘ CD IO V O‘
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fi v-n—«m oz CO V ‘GHO QD
ts oo cz fi cv 00 10 5 0 0 2 v c m m v
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0 2 0 2 09 00 00 m v v m uo CD (D b o b wCD Q fl CO AO
v-fl F 'l v-C
363TABLE XVI.
— 'I‘
RANSIT ION CURVE TABLE .
co»
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m v co oo v-c ODQD ONN O O v ao oo oo
1—1 v—l r-u-q r-NN N N CQ GO CO v v v ao no
C00
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toao
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XV I.
—TRANS IT ION CURVE TABLE. 367TABLE9
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TABLE XV II.— AREAS OF LEVEL SECT IONS.
S ide slopes to 1 .Base, 26 14 feet .
978 7
2771.
.0
.2
. 4
11 155
. 0
178 5
3465
1208 6
1490 6
Base, 26 15feet .. 1 . 2 . 3 . 4 .5
S ide slop es t o i .
.6
776 8
13855
1 130 1
374 TABLE XVII .
— AREAS OF LEVEL SECT IONS.
.0 F
flq
to
to
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t-nd
Hm
HHHHI—i
ca
m
tH—Qco
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m
mWk)
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wwwwwHHb-‘HH
i-n-n-n-u-t
Base , 25 16feet..0 . 1
Base,26 18 feet .
A) . 1
.2 . 3 .4 .5
Side slopes to 1 .
.6 . 7 .8 ».9
.2 .3 .4 .5
S ide slopes t o l .
.6 . 7
494 0
735 5
497 .
. 8 .9
XVIII.— AREA CORRECT IONS FOR THREE-LEVEL GROUND. 375
( SeeC orrec t ion ( km ho) 93 .
su m:SLOP ES 11 T o 1 .
2
46
8
1rd
Ow
Ov
10~
0
9
8
7
10s
75673 5
92
5
8l
08 .
8
11
1613
20
S IDE SLOPES 1 T o 1 .
sum :SLOPES 4 T O 1 .
. 4 .5.3km — ho
2
34
5
0
13
6
1
11
1
0
1
3
6
10
9
8
0
12
5
0
864
0
0
2
5
0
74
1
0
02
5
67
89
0
12
3
05
7667
111
1
1
1
1
1
0
5
1
8
65
5
6
112
2
3
4
5
6
765
4
3
2
1
0
94
w7
5
9
4
9
6
4
3
3
1
1
2
3
5
6
85
29
63
0
7
839
5
2
2
2
1
1
2
4
5
6
0T3SEPOLSEDIS
0hmh
376XIX .
— CUBIC YARDS PER 100 FEET . SLOPES g:1 .
Depth Base Ba se Base Base Base Base Base Base12 14 16 1 8 22 24 26 28
XIX.
— ~CUBIC YARDS PER 100 FEET . SLOPES 1 7} 1 .379
Dep th Base Ease Base Base Base Base Base Base12 14 16 1 8 20 28 30 32
1050 11161296 1593
1487 1813
1511 1600 168 91806 1902
2126
4296 4592
4628 49394970 5133 5296
566556896065 6250 6435
6644 68376050 7050
76748 109
9717
11467
1528 9
20117
21494
23961 2480522126 24704 25563
25457 263322311 1 26222 26067
TABLE XX . YARDS IN 100 FEET LENGTH . 38 3
Area Area . Area . 7 Area . Area .
1 Cub lc Cub lc Lub le Ouhlo Cub lcN 1. Yards , g‘g Yards. 253; Yards. fig; Yards. Yards.
. 32
. 33
. 34
. 35
. 36
. 37
. 39
.528
lkise‘
p er
Cent
.61
.62
.63
.64
.65
.66
.67
.68
.69
. 70
. 91
. 93
. 95
. 96AI»
0 l
. 98
. 99
Feet p er
33 - 264
51 744
Risep er
Cent .
90 U
. 24
. 25
. 26
. 27
. 28
. 29
. 30Mi—LHHH
HHH
uHHHH-au-‘Huu
Ln0)
Feet p erMi le .
Riseper
Cent
H
8
338 8
322
8
2
28
2
e
wwwwwwwww
8
8
963
8
8
8
8
8
8
XXL— RISE PER MILE OF VARIOUS GRADES.
Feet perMi le.
179 520
Incl
i
nati
on
.
Angle
of
TABLE XXII. —SLOPES FOR TOPOGRAPHY.
ertical
i
n1
00
Kor
i
zo
ori
zontal
Dl
stance
a
Rl
se
of
11
30
13
387
d o
.9 s u—l o q.‘“3
0 0 a Q:Q “0 0“c: “d “ 0 “
a
8 41 8 8 55315255; 2 3 £13 255
5 m 13:o Q os c u c s
TA BLE XXIII .— MATERIAL REQUIRED FOR ONE MILE OF TRACK .
RAILROAD SPIKES.RAIL WEIGHTS.
Pounds
per
Yard
.
gq
g
wa
a
wwwwuu
O
Q
MO
U‘
O
U‘
O
G
NJ
hort
Tons
2000
l
bs.
Long
Tons
2240
lbs
.
39 286
Si
ze
und
Head
.
"'o
rd
w3w3wwww°wW
W
M
M
X
X
X
X
Y
X
X
X
X
X
X
“P ”WWW!“
NUMBER OF CROSS-TIES.
D istance apart, 0 . t o c . , in Feet .
360400
450530
600680720
900
10001 1901240
1342
Requi red forT ies 2 ft . Apart .
pFor Ra i ls
£ 2 u go Weigh ingo <0
0
3 3 z
5870 45to 70 lbs.
5280 40 564690 35 40
3980 28 353520 24 35
31 10 20 302930 20 30
2350 16 252110 16 25
1770 16 20
1700 16 20
12 16i1570
NUMBER OF SPL ICE -JOINTS .
Two Bars with Four Bolts andNuts t o Each Jo int .
Length ofRai l in Feet .
TR'
IGO'
NoMET‘
RIC FORMULAS.
TABLE XXV II.
— TR1GONOMETRIC AND MISCELLANEOUS
FORMULAS.
T RIGONOMET RIC FORMULAS.
FIG . 98 . FIG . 99 .
In Fig 99 , let D C'E be the arc of a quadrant , ABC 9. right
triangle , the angle BA C sub tended by the arc CE A,and
consider the radius A C uni ty. T hen
Using the smal l letters a , b, c , to represent the s ides 0
right triangle in Fig. 9 8 or 99 , we m ay write
a bsm A = cosecA = sm A
cosA = secA = . . cosA .
b secA
tanA = . . tanA =
SOLUT ION OF TRIANGLES.
TABLE XXVII.— TRIGONOMETRIC AND MISCELLANEOUS
FORMULAS.
SOLUT ION OF RlGHT T RIANGLES .
Requi red. G iven. Form ulas.A , C ,
c a,b —
a) a
A, C ,
b a,c tanA = cot B = -Q
;c
C',b, c A , a C = 900 — A ; c = a cot A ; b = a cosecA .
C , a , c A , b C = 9OO — A ; a = b sinA ; e z b cosinA .
C , a , b A , c C = 90° — A ; a = c tanA ; b= c secA .
SOLUT ION OF OBLIQUE T RlANGLES.
Requi red. G iven Form ulas.
b A , B,a b
B A,a,b a
fi f454 4- 13) — 0 ) 1
/5 0
404 B) a , b, tanfl A B) Za b
tafi m B)a + b ’
— B)B = HA + m
— MA —
m
If sin4A = (E Mbc
cos4A'=
3“ a),
tan 1}A
A reaA rea A
,b, c A rea. 4bc sih
'
A
A rea A, B, c A rea. M
B)
GEN ERAL FORMULAS.
TABLE XXVII.— TRIGONOMETRIC AND MISCELLANEOUS
FORMULAS.
GENERAL FORMULAS.
sinA cos2A tanA 0 OSA .
sinA = 2 $ in 1}A 0 0 54 14 .
sinA1
cos2A ) .0 0 50 0 1 4
1 J l sin2A cotA sinAse0 1 4
cosA
cosA 1 2 sinHA 1 versA .
tanAsinA
~/ sec_
2A 1 .
0 0 3 1 1
tanAcos
‘Z A sin2A
0 OSA 1 COS2A
tan'
A1 1 cos2A
cot A sin2 A
cotA1 0 9 8 A
~ / cosec2A 1 .tanA sm A
sin2A 1 0 0 3 2 11cot A
1 8 111 2 11
the rec iprocal of any express ion for 0 0 511 .
cosec A the rec iprocal ofany express ion for sinA .
versA 1 cosA = 2 sin2 1}A .
verszi
0 0 3 1 4exsecA s
‘
ecA
5111 4 14
39 1 MISCELLANEOUS FORMULAS.
TABLE XXVIL— TRIGONOMETRIC AND MISCELLANEOUS
FORMULAS.
M ISC ELLANEOUS FORMULAS.
G iven. Fo rm ulas .
{ egular Polygon
Radius of base r
Slant heigh t s
Rad ius r,he ight h 2nrhRadius r 4
He ight h2 7rrh
Rad ius of its sphere r
Volum e ofPr ism or Cyl inder
A rea. of base bHeigh t 2 h
Pyram id or coneA 1 9 3 Of base bHelght r . h
Frus tum ofA f b b d b’ hPyram id or conerea O ases an
(b b’
~/ bb‘
Height h 3
Sphere Radius r 47579
Parallel s ides m and n pPerp . d ist . bet . them = p 2
Length of s ide = l 1 80°Numb er of s ides 2 n T nRad ius r m
“?
Sem i-axes= a and b‘
fl ab
Base 0,height h gbh
TABLE XXVIII .— SQUARE AND CUBE 110 0 1 5. 395
Squa re B oo t s and C ub e Ro o t s o f N u m b ers fro m .1 t o 2 8 .
No errors.
Cube. NO. Sq. Rt . Sq . Rt . C . R t .
396 TABLE XX IX .
—SQUARES,CUBES
, AND ROOTS.
T ABLE of Squa res . C ub es , Sq ua re R o o t s , and Cub e Boo ts .
o f N u m b ers fro m t o 1 0 0 0 .
REMARK OH T H E FOLLOWIN G TAB LE . Wherever the effec t ofa fifth decim al in the roots would beut dd 1 to the fourt h and final decim al in the tab le , t he addi tion has b een m ade. No erro rs .
Squa re. Cube. Sq . B t . Square. Cube. Sq . Rt . 0 . B t .
3721 22698 1
3844 23 8328
3969 250047
4096 262144 4 .
4 225 274625
4356 2874964489 300763
4624 314432
4761 328509
4900 343000
5041 357911
5184 373248
5329 389017
5476 405224
5625 4218 75 8 6603 4 2172
4 3267
8 28 1 753571
8464 778688
8649 804357
8836 83058 4
9025 857375 9 7468
12321 1367631
12544 1404928
1 2769 1 44 289 7
1 2996 1 48 1544
13225 1520875
13456 156089613689 160 1613
139 24 1643032
14161 1685159
14400 1728 000
?
Bl'i
fl
3151
35
“
398 TABLE XX IX .— SQUARES
,CUBES, AND ROOTS.
’
T AB LE o f Sq u a res , C ub es , S q u a r e Ro o t s , and C ub e R oo t s.o f N u m b ers fro m 1 t o 1 0 0 0 — (CONTINUED)
Square. Cube. Sq . R t . N o . Square. Cube. Sq . B t . C . R t .
6300 1 158 13251 316 99856 3155449663504 1600300 8 31 7 100489 31855013
64009 16194277 318 101 124 3215743 264516 16387064 319 101 761 32461 759
65025 1658 1375 320 102400 32768000 1 7 8885 68399
65536 16777216 16.
66049 1697459366564 1 717351 26708 1 1 7373979
67600 1 7576000 161 245
6 8894
336 1 1 2896 37933056337 1 13569 38272753338 1 14244 38614472339 1 1 4921 38958219340 1 15600 39304000
341 1 16281 39651821342 1 16964 4000 1688343 1 1 7649 40353607344 1 18336 40 707584345 1 19025 41063625
346 1 19716 41421736347 1 20409 41 78 1923348 121 104 4214 4192
349 1 21 80 1 42508549350 122500 428 75000 18 7083
8 1796 233936568 2369 23639903
8 2944 2388 78 72
83521 24137569 1 7 .
84 100 24389000
8468 1 24642171
85264 248 97088
85849 25153757
86436 2541 2184
87025 256723 75
8 7616 25934336
88 209 261980 73
8880 4 26463592
8940 1 26730899
90000 27000000
90601 27270901
9 1204 27543608
91809 2781 8 127
92416 28094464
93025 28372625
93636 28652616 371 137641 510648 1194249 28 934443 372 138384 514788 4894864 29218 1 1 2 373 139 129 518951 1 79548 1 29503629 374 1398 76 5231362496100 29 791000 375 140625 52734375
96721 30080231 376 141376 5315737697344 303 71328 377 142129 5358263397969 3066-429 7 378 14288 4 540 10152
98596 30959144 379 1 4364 1 54439939
99236 31255875 380 144400 548 72000
TABLE XXIX .—SQUARES, CUB ES, AN D ROOTS. 399
T ABLE o f Squ a res , C u b es , Sq ua re R o o t s , and Cub e R oo t s .
N o .
441
442
Squa re.
19448 1195364196249197136198025
o f N u m b ers fr o m 1 t o 1 0 0 0 — (CONTINUED )
Cube.
857661218635088 8869383078 7528384
88 121125
Sq . Rt .
21
21 0238
21 0476
N o.
446447
448
449
450
451
452
453
454
455
456457458
459
460
461
462
463464465
466467468469470
Square.
198916199 809
200704
201601202500
203401
204304
205209
2061 16207025
2079362088 49
20976421068 1
211600
212521213444
214369
215296216225
217156218089219024
219961
244036
8871653689314623
8991539290518849
9 1 1 25000
917338519 2345408
929596779357666494196375
94818816
9607191296702579
97336000
9797218 1
98611 128
9925284799897344
100544625
101194696101847563
102503232
103161709103823000
( l . 11 t .
400 T ABLE XXIX — SQUARES, CUBES, AND ROOTS .
T AB L E o f S u
559
303601304704
305809
306916308025
309 136310249
31 136431248 1
313600
314721315844316969318096319225
320356321489
322624
323761
324900
3260418 27184
328329
329476
330625
167284151168 1966081691 123771 70031464
1 70953875
1 718796161 728086931 73741 1 1 2
1746768 791 75616000
17655848 11 775043 281 78453547
1 79406144
180362125
1 813214961 8 2284263183250432
1 84220009
1 85193000
1 8616941 11 8 7149248
1 88 1325171 89 1 19224
190109375
8 .2670
61661 7618619620
621622623624625
626627628629630
631
633
635
636637
639
37945638068938 1924
383161
384400
3856413868 84388 129
389376
390625
391876393129
394384
395641396900
398 161399424
400689401956403225
40449640576940 7044
408321
409600
233744896234885113
23602903223 7176659238328000
239483061240641848241804367242970624
244 140625
24531437624649 1883247673152248858 189
250047000
251239591
2524359682536361372548 4 0104
256047875
257259456258474853
25969407 22609171 19262144000
25.
a res , Cub es , Sq u a r e Ro o t s , a n d C u b e R o o m ,
0 11Nu m b ers fro m 1 to 1 0 0 0 - (CONTINUED )
402 TABLE XX IX .- SQUARES, CUB ES
,AN D ROOTS.
T AB LE o f Squ a res , C ub es , Sq ua r e R oo t s . a nd C ub a R o o t s ,
O f N u m b ers fr o m 1 t o 1 0 0 0
774
775
779
790
79 4
795
796
797
798
80 1
802
803
8 04
806
809
8 10
8 11
8 12
8 13
8 14
8 15
8 168 17
8 18
8 19
8 20
8 21
8 22
8 25
8 268 27
8 28
829
830
831
832
833
834
Squa re.
594 44 1595984597529599076600625
602 176603 72960528460684 1608 400
60996161 1524
613089
614656616225
61 7796619369
620944
622521
624 100
62568 1627264628849630436632025
633616635209
6368 04638 401
640000
64160164 3204
644 809
6464 16648025
649636651 249652864
65448 1656100
657721659344660969662596664225
66585666748 9669 124670761
672400
67404167568467732967897668 0625
6822766839 29
685584
68 7241
688900
690561
692224
69388 9
695556
697225
Cube.
45831401 146009964846188 99 1 746368 48 24
46548 4375
4672885764690974334709 10952
4 72729 139
4 74552000
47637954 14 78 211 7684 8004868 7
48 1890304
483736625
48558765648 74434034 893038 72
491 169069493039000
494913671496793088498677257500566184
5024598 75
504358336
506261573
508 169592
51008 2390
51 2000000
513922401
5158 49608
51778 1627
519 7 18 464
521660 125
523606616525557943
5275141 1 2
529475129
53144 1000
53341 1731
53538 7328
537367797539353 144
541343375
543338496
545338513
547343432
549353259
551368000
553387661
5554 1 2248
55744 1 767559476224
561515625
5635599 76565609 283
567663552
569722789
571 78 7000
573856191
575930368
578009537
580093704
58 21 8 2875
Sq . R t .
2719464
28 .
28 6531
28 7054
28 14944
28 8617
N o .
836837
838 .
839
840
841
8 4 2
8 43
8 44
8 45
846
8 4 7
848
84 9
850
85
852
853
854
855
8718 72
8 73
8 74
8 75
8 76
8 77
8 78
8 79
880
88 1
8 8 3
8 8 4
8 85
886
88 7
88 8
88 9
8 90
8918 9 2
8 93
894
895
8 968 97
8 98
899
900
Square.
698896700569702244
703921
705600
70728 1
708964
7 10649
712336
714025
71571671 7409
719104
72080 1
722500
724201
725904
727609
729316
731025
75864176038 47621 29
7638 76
765625
767376769 129
77088 4
77264 1
7744 00
7761617 77924
77968 978 1456783225
784996786769
788544
790321
792100
793881
7956647974 49
799 236801025
8028 16804609
8 06404
808 20 1
8 10000
C ube.
584277056586376253588 4 80 472
590589719
592704000
594823321596947688599077 10760 1211584603351 1 25
60549573660764542360980019261 1960049614125000
6162950516184 70208620650477622835864625026375
66077631 1663054848
66533861 7667627624
6699218 75
6722213766745261336768 36152
679 151439
68 1472000
68379784 1686128968
68846538 7690807104
693154 125
695506456697864103
700227072
702595369
704969000
70734 7971
709 732288
7121219577 14516984
71691 7375
719323136721734273
724 150792
726572699
729000000
Sq . R t .
29 .
551 0345
251 0689
29 1 719
29 1890
521 5296551546658 1 5635551 5804
29 597329 6142
29 631 1
21 8496
2928664
151 883 1
551 899 8
29£HG6
551 9333
29A¥fl¥3
30 .
9 4912
£1 553751 5574
51 5647
9 5756
£1 58 28
51 5865
9 .5937
$L6010
116190
£163 70
£16406£1 6442
£164 77£16513
TABL E XX IX .
— SQUARES, CUBES, AN D ROOTS . 403
T AB L E o f Sq u a res , C u b es , S q ua re R o o t s , an d C ub e R o o t s ,o f N u m b ers fro m 1 t o 1 0 0 0 — (Com m u n )
Squa re. Cube. Sq . R t . N o . Squa re. Cube . Sq . R t . C . Rt .
81 1 80 1 73 143270 1 951 904401 860085351
8 13604 7338 70808 952 906304 862301408
8 15409 736314327 953 908 209 865523177
8 17216 738 763264 954 9 10116 868 250664
905 8 19025 741217625 955 9 12025 8 709838 75
906 8 20836 743677416 956 9 13936 8 737228 168 22649 746142643 957 9 158 49 8 76467493
8 24464 748613312 958 9 17764 8 7921 7912
8 2628 1 75108 9 429 959 9 1968 1 88 1974079
8 28 100 753571000 960 9 21600 8 8 4736000
8 29921 75605803] 961 923521 887503681831744 758550528 962 925444 890277128
9 13 8 33569 761048 497 963 927369 893056347 31 .0322
835396 763551944 964 929296 8958 41344 31 .0483
9 15 8 37225 7660608 75 965 931225 898632125
768575296 966 933156 901428696 31 .08058 408 89 77 1095213 967 935089 9042310638 42724 773620632 968 937024 9070392328 44561 776151559 969 938961 9098532098 46400 778688000 970 940900 91 2673000
8 48 241 78 122996] 971 942841 91549861 1 31 .1609850084 783777448 972 94478 4 9 18330048851929 786330467 973 946729 921 167317853776 788 889024 9 74 948676 924010424 31 .2090855625 791453125 9 75 950625 926859375
857476 794022776 976 952576 929714176 31 .2410859329 796597983 977 954529 932574833861 184 799 178 752 978 956484 9354413528630 41 80 1 765089 9 79 95844 1 938313739 31 .2890864900 804357000 980 960400 941 192000
866761 80695449] 98 1 962361 9440761 4]868624 8 09557568 98 2 964324 9469661688 7048 9 8 12166237 983 966289 94986208 7 31 .35288 72356 8 14780504 98 4 968 256 952763904
935 8 74225 8 1 7400375 985 970225 955671625
8 76096 8 20025856 986 972196 9585852568 77969 822656953 987 974 169 961504803 31 .41668 798 44 8 25293672 98 8 976144 9644302728 8 1721 8 27936019 989 978 121 967361669883600 83058 4000 990 980100 970299000
8 8548 1 833237621 991 98208 1 97324227188 7364 8 358968 8 8 992 98 4064 9761914888 89249 8385618 07 993 986049 979 1466578 9 1 136 8 4123 238 4 994 98 8036 98 2107784 31 .5278893025 8 43908625 30 . 7409 995 990025 9850748 75
946 894916 8 46590536 996 992016 98 8047936 31 .55959 47 8 968 09 8 49278 1 23 30 . 7734 99 7 994009 99 10269739 48 89 8 704 851971392 998 996004 99401 1 992 31 .591 19 49 900601 854670349 999 998001 997002999 31 .6070950 902500 857375000 1 000000 10000000001000
'
1 0 fi n d t h e sq ua re o r c u b e o f a ny w h o l e n u m b e r en d ingw i t h c i p h e r s . F i rst , ow l t. all t he final c iphers. Take from the tab le thesquare or cube ( an t he case m ay be) of the res t of t he num ber. T o this square add twice as m anyc iphers as t here were fina l c iphers in t he original num ber. T o t he cube add three t i m es asin t he original num ber. T hus , for 90500 2 ; 9052 2 8 190 25. Add twwe 2 C i p hers . o b taim ug 8 1.
-30250000.For 90500 3 , 9053 2 74 12 17625. Add 3 tunes 2 c iphers , ob ta ining 74 1217625000000 .
