R PA P P A 
 
 777 
 
 
MATHEMATICS ?,!5 
 PAPER LOCATION 
 OF A RAILROAD 
 
 J, C. L. FISH 
 
MATHEMATICS 
 
 OF THE 
 
 PAPER LOCATION 
 OF A RAILROAD 
 
 J. C. L. FISH 
 
 Assoc. M. Am. 8oc. C. E. 
 
 Associate Professor of Civil Engineering 
 Leland Stanford, Jr., University, Palo Alto, Cal. 
 
 NEW YOKK 
 
 M. C. OLAEK 
 
 13-21 PARK Row 
 
 1905 
 
Copyright, 1905, by M. C. Clark 
 
The object of this article is to present the 
 mathematical steps involved in preparing, from 
 the paper location of a railroad, a set of mathe- 
 matically consistent alinement notes by which to 
 run the corresponding field location; and to sug- 
 gest an orderly arrangement for the necessary 
 computations. 
 
 Mr. W. B. Storey, Jr., in discussing a paper by 
 Michael L. Lynch on Railroad Location, says:* 
 
 On the Southern Pacific System the location ia trans- 
 ferred from the paper to the ground, not by scaling, but 
 by calculation. Each tangent is fixed, and the connect- 
 ing curves are all calculated in the office by carrying the 
 line from one fixed tangent around through the prelim- 
 inary to the next tangent. These calculated notes are 
 then given to a machine known as the locating party 
 and put on, the ground by it. 
 
 The writer does not remember having seen else- 
 where any reference to the calculation of field 
 notes; but in various articles on railroad location 
 he has noted direct or implied reference to scal- 
 ing. 
 
 To make clear what is meant by scaling and by 
 calculating, and to show the advantage of calcu- 
 lating over scaling, let us find, by the two 
 methods, the station and "plus" via the prelimi- 
 nary and via the location for the check point A, 
 Fig. 1, the first crossing of the location and pre- 
 liminary w;hich have a common starting point, Pi. 
 
 BY SCALING. The angle at Pi, between pre- 
 liminary and location, is scaled in the course of 
 preparing the notes for field location. Simply for 
 the purpose of getting a check on the location at 
 
 Trans. Am. Soc. C. E., Vol. XXXI., p. 92. 
 
 (Reprinted from Engineering News, 
 March 16, 1905.) 
 
 381115 
 
5000 FT. 
 
 4000 (1500,4000) R 
 
 1000 
 
 Pape 
 
 (3000, 4200) fg 
 
 <fr2f.' /r 
 
 P 7 (4000, 4-500) 
 
 4000 FT. 
 
 FIG. 1. METHOD OF PLOTTING PAPER LOCATION OF A 
 RAILWAY LINE. 
 
the point A we scale the distance PiA on the loca- 
 tion and the distance PaA on the preliminary, thus 
 obtaining the station and plus for A via each of 
 the two lines. We now have of the triangle 
 PiPaA the given side PiPa (taken from the field 
 notes of preliminary), the given angle PiPaA, the 
 two scaled sides PiA and PaA, and one scaled 
 angle, APiPa. All the scaled quantities are af- 
 fected 'by the errors of scaling, which are large 
 compared with field errors, and it is evident that 
 our values for the five parts of the triangle are 
 mathematically inconsistent, since any three parts 
 (which include one side) of a triangle determine 
 the other parts. The result is that when the field 
 party has located to A, and found they do not 
 check on the preliminary within several feet, they 
 have no means of telling how much of the error is 
 due to surveying and how much to scaling. In 
 this case the surveying checks the scaling, but 
 there practically is no check on the field work. 
 
 BY CALCULATION. iBy this method we scale 
 only so many parts of the triangle PiPaA as will, 
 together with the given part or parts, determine 
 the size and shape of the triangle. Side PiPa and 
 angle PiPaA are known from the preliminary 
 notes. If, then, we scale the angle PaPiA we shall 
 have values for three parts of the triangle, from 
 which we can compute the sides PiA and PaA. 
 These five parts of the triangle 'two given, one 
 scaled and two computed 'are mathematically 
 consistent; and when the location party arrives at 
 A and find they do not exactly check on the pre- 
 liminary, they know that the error is all charge- 
 able to field work (assuming that no errors have 
 been made in computing). In this case the field 
 work is really checked. Of course the error may 
 be in running the preliminary or the location, or 
 more probably in both. The check is on the sur- 
 veying on both lines between Pi and A. If the 
 error is not within the limit of error permitted, 
 the location is re-run (using the same notes), and 
 if this is found to be practically correct, the next 
 step is to re-run the preliminary from Pi to A. If 
 the error is not discovered here it must be found 
 in the office work computations or copying of 
 notes. It cannot be due to scaling. 
 
 5 
 
MATHEMATICS OF PAPER LOCATION: 
 USING RECTANGULAR CO-ORDINATES. The 
 chief points of the preliminary, surveyed by any 
 method or combination of methods, have been 
 plotted by rectangular co-ordinates, and our map 
 shows points Pi, P 2 , P 3 , P4, Ps, Pe and PT. The co- 
 ordinates of each point are written by the point, 
 the value of the abscissa first. The engineer has 
 drawn in pencil on the map the location Pi, PCi, 
 PTi, PCs, PT 2 , PCs. PCi and PTi are respectively 
 the beginning and end of a 6-degree curve; and 
 PCz and PTa are the beginning and end respec- 
 tively of an 8-degree curve. PCs is the beginning 
 of a curve, and is considered the end of this loca- 
 tion. It will be noticed that each curve is num- 
 bered, and that each letter standing for a curve 
 element is given for a subscript the number of 
 its tmrve; e. g., the central angle of curve No. 2 is 
 AS. All dimensions which appear on the map at 
 this time are put in parentheses on our figure in 
 order to distinguish them from the quantities 
 computed in the work of getting out the location 
 notes, which are written without parentheses. 
 We have nothing to do here with the question of 
 the proper place for the location line on the map: 
 that is an engineering problem, while our work is 
 to take the location line as we find it, and get out 
 the notes for transferring it to the ground. Hav- 
 ing no need of the topography in what follows, It 
 has been omitted from our map. The distances 
 have been computed here to the nearest foot only, 
 although it is customary to consider tenths of a 
 foot. Of course, in any given location the limit of 
 error in field work should control the precision of 
 computation. To save space there have been 
 omitted from Fig. 1 the station numbering of the 
 preliminary and of the location. 
 
 1. PRODUCE THE TANGENTS to intersect 
 at Vi and V 2 . 
 
 2. SCALE THE CO-ORDINATES of the initial 
 point, the final point, and of all the points of tan- 
 gent intersection. In our case, the initial point of 
 the location is at the origin of co-ordinates, and 
 coincides with the initial point of the preliminary. 
 The co-ordinates of Vi and V 2 and PCs are written 
 by the points on the map as they are scaled off, 
 x first and y second. 
 
 6 
 
No further scaling is done, except that every 
 computed distance and angle should be roughly 
 checked by scaling. We are now ready to take up 
 the computing. 
 
 3. COMPUTE THE BEARINGS AND 
 LENGTHS OF TANGENTS PRODUCED TO 
 INTERSECTION. The plus direction of the 
 y-axis is taken as north. The tangent of the 
 bearing of PiVi is 
 
 tan *S PV = X Y ,-XP, 3,zso~o 
 
 (Throughout, the subscript of an x or a y of any 
 point is the symbol which designates that point; 
 and the subscript of is composed of the two 
 symbols which designate the terminal points of a 
 line.) 
 
 log 3,250 3.51188 log 3,250=3.51188 
 
 log 1,610 = 3.20683 log sin 63 39' = 9.95236 
 
 log tan 63 39' = 0.30505 log 3,627 = 3.65952 
 
 From the logarithmic computation we find the 
 bearing and length of PiVi to be N 63* 39' E, 
 3,627 ft. 
 The tangent of the bearing of ViVa is 
 
 , 3,640-3,250 2,030 
 
 and 
 
 log 2,100 = 3.32222 n log 2,100 = 3.32222 
 log 2,080 = 3.30750 log sin 46^ 58' - 9.85609 
 
 45 58' = 0.01472 n 2,921 = 3.46553 
 
 That is, the bearing and length of ViVi are 
 N 45 58' W, 2,921 ft. (The negative value of 
 the tangent shows that the bearing is either S E 
 or N W, but the map indicates that the bearing 
 is NW.) 
 
The tangent of the bearing of VaPCs is 
 
 and 
 
 log 2,380 = 3.37658 logr 2,380 3.37658 
 
 log 480 = 2.68124 log sin 78 36' = 9.99135 
 
 log tan 78 36' = 0.69534 log 2,428=3.38523 
 
 giving- for VaPCa the bearing and length of 
 N 78 36' E, 2,428 ft. 
 
 We now scale the map for a rough check on 
 these computed bearings and distances. 
 
 4. COMPUTE CENTRAL, ANGLES OF 
 CURVES. It is evident from the map that the 
 central angle for curve No. 1 is 
 
 Ai = ^ P]VI + ^v!V 2 = 63 39 ' + 45 58/ = 109 37/ ; 
 and the central angle for curve No. 2 is 
 
 Aa = ^Vj + ^V 2 PC 3 = 45 58 ' + 78 36/ = 124 34/ - 
 We roughly check these values by scaling the 
 map with the protractor. 
 
 5. COMPUTE TANGENT - DISTANCES OP 
 CURVES. Using a "table of tangent-distances 
 for a 1. curve" we find the tangent-distance for 
 curve No. 1 is 
 
 Ti = 1,355; 
 and for curve No. 2, 
 
 T 2 = 1,364. 
 
 We obtain a rough check on these by scaling. 
 If in any case the value of A is beyond the 
 limits of the table, of course the tangent-distance 
 must be computed from the formula: 
 T = R tan A/2. 
 
 6. COMPUTE CURVE LENGTHS. Length of 
 curve is equal to one hundred times the ratio of 
 central angle to degree of curve. 
 
 Length of curve No. 1 is 
 
 Lj 100 Ai / I>i = 100 (109 370/6 = 100 (109.62) / 6 
 = 1,827. 
 
 8 
 
Length of curve No. 2 is 
 
 L s 100 Aa/D a 100 (124 34') / 8 = 1OO (124.57) / 8 
 1,557. 
 
 We now step off the curve lengths on the map 
 to obtain a rough check on the computed values. 
 
 7. COMPUTE LENGTHS OF CENTER-LINE 
 TANGENTS. The length of the first tangent, as 
 the map shows, is 
 
 PiPCi = PiVi Ti = 3,627 1,355 = 2,272. 
 The second tangent is 
 
 PTi PC 2 = ViVs (Ti + T 2 ) = 2,921 (1,355 + 
 1,364) = 202; 
 And the third tangent is 
 
 PTa PCs = Va PCs T 2 = 2,428 1,364 = 1,064. 
 The rough check by scaling is now employed. 
 
 8. COMPUTE STATION AND PLUS FOR 
 CURVE AND TANGENT POINTS. The station 
 and plus for Pi is + 00. 
 
 The station and plus for PCi Is PiPCi = 
 (22 + 72). 
 
 The station and plus for PTi is (22 + 72) + La. = 
 (22 + 72) + 1,827 = (40 + 99). 
 
 The station and plus for PCs is (40 + 99) + PTi 
 PCs = (40 + 99) + 202 = (43 + 01). 
 
 The station and plus for PTa is (43 + 01) + L = 
 (43 + 01) + 1,557 = (58 + 58). 
 
 The station and plus for PCs is (58 + 58) + PT 
 PCs = (58 + 58) + 1,064 = (69 + 22). 
 
 Now we prick off the stations on the map, thus 
 checking roughly these values. 
 
 9. COMPUTE ELEMENTS FOR CHECK 
 POINTS. On our map the location crosses the 
 preliminary at the point A, which we use as a 
 check point. There is no corresponding point at 
 the other end of the line, and, in order to obtain 
 a check, we draw the line PCsP? and compute its 
 bearing and length, to be run in the field as an 
 auxiliary line to check on point PT. 
 
 CHECK POINT A. We first write the equa- 
 tions for lines, P 2 Ps and PiVi. The general equa- 
 tion for a straight line is y = ax + b, 
 
 where a = (y n y^ / (x n x k ) 
 
 and b = y k ax k . 
 
 (The subscripts k and n refer to the initial and 
 final points respectively of any line.) 
 
 9 
 
For line P g P 3 a = (y 3 y 2 ) / (x 3 x 2 ) (1,700 600) 
 (2,600 1,600) = 1,100/1,000 = 1.1; and b = y, ax 2 = 
 600 1.1 (1,600) = 1,160. Equation of line P a P 8 is 
 therefore, 
 
 y=l.l x + (1,160) 
 or y - 1.1 x 1,160. 
 
 For line PiVi 
 
 i, 6W0 _ 04.954 
 
 3,250-0 
 
 and 
 
 making the equation of PiVj 
 
 y = 0.4954 x. 
 
 We now compute the co-ordinates of the point A 
 of intersection of PaP2 and PiVi. 
 
 If the equation of the first line be written 
 
 y = ax + b 
 and the equation of the second be written 
 
 y = a' x + b' 
 then the co-ordinates of this common point are : 
 
 and a check is had in the equation 
 
 y ~a!x> +V 
 
 Substituting numerical values of a, b, a', b', for rou 
 point A, we get : 
 
 (- J.160J - 951 
 We scale the map to get a rough check on the computations. 
 
 It remains to find for A the station and plus 
 via each line. 
 The distance 
 
 log 319 = 2.50379 
 log sin 42 16' = 9.82775 
 
 log 474.8 = 2.67604 
 The logarithmic computation makes P 2 A 474.3. 
 
 10 
 
The station and plus of A on the preliminary Is 
 
 PI P 2 -f ?2 A. = 1,709 + 474.3 - 21 + 83.3. 
 The distance 
 
 as computed by logarithms here : 
 
 log 1,919 = 3.28299 
 log sin 63 39' = 9.95236 
 
 log 2,141 = 3.330Q3 
 
 The station and plus for A on the location is, 
 then, 21 + 41. 
 
 Scale the map to roughly check these values. 
 
 It is evident that station 21 + 41 on the location 
 survey should coincide with station 21 + 83.3 on 
 the preliminary. 
 
 In cases in practice a check point may be con- 
 veniently obtained by producing a location tan- 
 gent to intersect the preliminary, and making the 
 computation in the foregoing manner. 
 
 CHECK LINE PCaF?. When a crossing of the 
 location with the preliminary is not near at hand 
 for a. desired check on the field work, a check 
 line, or tie line, is drawn between a chosen point 
 of the location and a chosen point of the pre- 
 liminary; and the bearing and length of the check 
 line are computed. As an example, draw the 
 check line PCsP?, and find the bearing by the 
 equation: 
 
 Vr, -ypc 3 4,500-4,120 
 and find the length 
 
 log 470 2.67210 log 470 2.67210 
 
 log 380 = 2.57978 log sin 51 03' = 9.89081 
 
 log tan 51o 03' = 0.09232 log 604.4 = 2.78129 
 
 We find the bearing and length of PCaP? to be 
 N 51 03' E, 604.4 ft. The location having been 
 carried in the field to the point PCs, the transit- 
 man deflects to the left at this point the angle 
 
 ^PCs - ^PCsPr = 78 36 ' ~ 51 3 ' ---- 27 33/ ' 
 and the chainmen lay off 604.4 on this course, and 
 
 11 
 
should by so doing- arrive precisely at PT on the 
 preliminary. 
 
 10. COMPUTE CURVE DEFLECTIONS. PCi is 
 22 + 72 making- the first sub-chord on the 6-degree 
 curve, 28 ft. The corresponding deflection is .28 x 
 3 = .28 x 180' == 50', i. e., the tangent deflection 
 for station 23 is 50 minutes. The deflection for 
 sub-chord at .PCa is .99 x 240' = 238' = 3 58'. 
 The deflection for sub-chord at PTi is .99 x 180' = 
 178' = 2 58'. The deflection for sub-chord at 
 PT 8 is .58 x 24tf = 139' = 2 19'. 
 
 ALGORITHMS. To facilitate explanation the 
 foregoing computations have been put down in a 
 
 Red- 
 
 
 Fig. 2. Design for Rulings on Computing Paper. 
 
 rambling manner which makes the computed 
 quantities hard to find when wanted. For a lon- 
 ger location than this, economy of time and effort 
 requires that the routine computations be syste- 
 matically arranged, that like operations as well as 
 like quantities may be brought together. By this 
 means entering the data and making the computa- 
 tions in a short time become largely mechanical 
 processes, and at the end the computed quantities 
 stand in tabular order and may be quickly found 
 when wanted. For the computations of this ar- 
 
tide the following computation forms, or algo- 
 rithms, are suggested. While these may not pre- 
 sent the best arrangement, they will at least show 
 the advantage of order over disorder. The reader 
 may be interested in the fact that the writer's stu- 
 dents buy for their computing, letter-size sheets 
 of paper ruled on one side with the special design 
 shown in Fig. 2. By the use of this paper any 
 algorithm may be followed without drawing lines, 
 and columns of digits are kept vertical automat- 
 ically. 
 
 Algorithm 1 is for computing the bearing and 
 length of lines which are terminated by points of 
 known co-ordinates. Column A contains the num- 
 bers of the horizontal lines of the algorithm. Col- 
 umn B contains the symbols. Xk, yk designate the 
 co-ordinates of the initial point of any line; x n , y n 
 designate the co-ordinates of the final point of the 
 line; )3kn is the bearing, and dk n is the length of 
 the line. The lower half of this algorithm has to 
 do with the check equation: 
 
 hypot. = (base 2 + altitude 2 ) 1 / 2 . 
 
 Notice that the values on line 21 check those on 
 line 12. Of course this check will not detect errors 
 made in entering values of co-ordinates on lines 
 1, 2, 3, 4. 
 
 When a table of squares is at hand it may well 
 be used in place of logarithms for the computation 
 of checks. Columns C, D, E, F give for the line 
 Pi Vi, Vi Va, Va PCs, and PCsP? respectively, the 
 numerical quantities corresponding to the symbols 
 of Column B. It is suggested that the order of 
 steps in computing be: (1) Enter the co-ordinates 
 for all the lines concerned; (2) Set down on lines 
 5 and 6 the differences; (3) Enter all the logs on 
 line 7; (4) Enter all the logs on line 8, and so on; 
 every operation in Column C being immediately 
 repeated for the succeeding columns. 
 
 Algorithm 2: Column A contains the numbers 
 which have been used to designate the curves. In 
 B will be found the degree of each curve. Each 
 degree of curve is repeated in parenthesis, with 
 the minutes expressed in decimal of a degree. In 
 C are entered the central angles, each of which 
 has been computed from two values of taken 
 from Algorithm 1. For example: C2 (i. e., the 
 
 13 
 
ICC*'* 
 
 CO COO 
 
 _, ^ibw^wS ^ 
 
 ^ CO T-J iM iH CO kO l> 
 
 ko kOko koc* 
 
 2s: 
 
 OoOo 
 
 rHOkOr-iCO.SlOCS CO CO<N-* CD rHTjt 
 >> H-^OS-^iMCD'H k(5 iHkOiM CO COC^ 
 <D l^ 05 0500 
 
 CO'M'O'OS'CO' t-coio 
 
 l O O C<l 
 
 t- 
 
 ^ OOOOOO L- ^^^ 
 
 O7 kO rH kO r-l 00 CO kC CD Cq (M - CD CO O O CS ' 
 
 iMCD<NCDOOOO O COkCCO ^5 t-DCDC<lkOo* 
 
 CO r-l CO rH rH CD ifl C^ 05 CO CO CO CO kO 05 iH 05 05 CO 
 
 rHO O kOkO Q C* rH O kO CO r-l kO 
 
 eoco o' oico *co t-co 
 
 s ss 
 
 , 
 
 be M 6C tfi 6C 
 00 O 00 
 
 II .t 
 
 : : : :3 
 
 rH <N CO "* kO CD t- 00 05 OrH(M 
 
 14 
 
-9p paoqoqng 
 
 PITB "BIS 
 
 snid pire " 
 
 iu -i . . . , 
 
 o .to : : : M 
 
 01 <S1 . . 
 
 Z jo urns 
 
 jstp -Sum I S 
 
 H -UOT!J09e.T9!lUI 
 
 g 05 p9onpojd ^ c^ 
 
 4 (ITTQ^TTTJI CD 
 
 iS i i| : 
 
 3" 
 
 9AJUO 
 
 :g:: s ::: 
 ifi : '% : ' : : 
 
 <] r 
 
 15 
 
quantity in Column C and on line (2) is 109 37', 
 which is the sum of C13 (Alg. 1) and D13 (Alg. 1). 
 C3 and C6 give the central angle with minutes re- 
 duced to decimal of a degree. D contains the 
 tangent distance for each curve. E contains the 
 curve lengths. In P are entered the lengths of 
 tangents produced to intersection points, copied 
 from line 12 of Alg. 1. G contains the amounts to 
 be subtracted from quantities in F to obtain the 
 quantities in H. For example: G4 is the sum of 
 D2 and D5, and H4 is F4 less G4. I contains the 
 station and plus for each point of curve; and J 
 contains the station and plus for each point of 
 
 ALGORITHM III. Line Equations. 
 
 Formulas : a = (y n 
 
 = y k 
 
 x n 
 
 Line P 2 P 3 Line PiVj 
 
 1600 
 
 600 5 
 
 2600 3250 
 
 4 y n 1700 1610 
 
 5 y n y k 1100 1610 
 
 6 x n x k 1000 3250 
 
 7 log(y n -y k ) 3.04139 3.20683 
 
 8 log(x n x k ) 3.00000 3.51188 
 
 9 log a 0.04139 9.69495 
 
 10 a 1.1 0.4954 
 
 11 logx k 3.20412 
 
 12 logax k 3.24551 
 
 13 ax k 1760 
 
 14 b 1160 
 
 15 (Eq.)y = ax + b y = l.lx 1160 y = 0.4954x + 
 
 ABC 
 
 ALGORITHM IV. Co-ordinates of Intersection Point. 
 
 Formulas : x = (b' b) / (a a') ; y = ax + b ; 
 check, y = a'x + b'. 
 
 Check Point A. 
 
 1..., (Eq.iy = ax + b y=l.lx 116O 
 
 2 (Eq.)'y = a'x + b' y= 0.4954 + 
 
 3 a 1.1 
 
 4 a' 0.4954 
 
 5 b 1160 
 
 6 b' 
 
 7 b' b 1160 
 
 8 B a' 0.6046 
 
 9 log(b b) 3.06446 
 
 10 log (a a' 9.78147 
 
 11 log x 3.28299 
 
 12 log a 0.04139 
 
 13 log a' 9.69495 
 
 14 log ax 3.32438 
 
 15 log a'x 2.97794 
 
 16 ax 2110. 
 
 17... a'x 950.5 
 
 18 x 1919. 
 
 19 y(=ax + b) 950. 
 
 20 y(=a'x + b') 950.5 
 
 1G 
 
tangent. 12 is rewritten from HI; J2 is 12 plus 
 E2; 15 is J2 plus H4; J5 is 15 plus E5, and so on. 
 K and L contain the subchord deflections. 
 Algorithms 3 and 4 will need no explanation. 
 
 11. ALINEMENT NOTES FOR LOCATION. 
 We are now ready to enter the results of our com- 
 putations in the field book in the usual form of 
 alinement notes. 
 
 Location Alinement Notes. 
 Computed 
 
 Sta. Curve Point. Deflections. Bearings. 
 69 + 22 PC 8 Check : Deflect 27 33' (bearing N 51 
 
 03' E) to left and run 604/4 to P 7 which 
 .... is Sta. 84 + 26 on preliminary. 
 
 60 
 
 59 N 78 36' E 
 
 58 + 58 PT 2 62 17' N 16 20' E 
 
 58 59 58' 
 
 57 55 58' 
 
 45 7 58' 
 
 44 3 58' 
 
 43 + 01 PC 2 8 R 00' N 45 58' W 
 
 43 
 
 42 
 
 41 N 46 58' W 
 
 40 + 99 PT! 54 48.5' N 8 51' E 
 
 40 51 50' 
 
 39 48 50' 
 
 25 6 50' 
 
 24 3 50' 
 
 23 50' 
 
 22 + 72 PCi 6 L 00' N 63 39 E 
 
 22 
 
 21 + 41 Check : This point is 21 + 83.3 on preliminary. 
 
 21 
 
 "2" 
 
 1 
 
 This is Sta. of preliminary. For first location 
 
 course deflect 5 48' to left of first preliminary 
 
 course . 
 
 THE MATHEMATICS OF PAGPER LOCATION : 
 USING POLAR CO-ORDINATES.-^To compute 
 the location field notes without employing rect- 
 angular co-ordinates, proceed thus: 
 
 1. The location having been laid down on the 
 map as shown, scale PiVi, ViV2, "WFCa, and with 
 the protractor scale the bearings of these three 
 lines. 
 
 2. OomDUte the central angles A i and . 
 8. Compute curve lengths La. and L 2 . 
 
 4. Compute tangent distances Ti and Tz. 
 5. Compute lengths of tangents PiPCi, PTi PCs 
 and PTaPCs. 
 
 17 
 
6. Write station and plus for Pi, PCi, PTi, PC, 
 PT a , and PC 8 . 
 
 7. Find preliminary station and plus, and loca- 
 tion station and plus, for check point A, thus: (a) 
 Compute angles of triangle PilPsA; (b) Prom the 
 known side and angles of this triangle compute 
 the sides PiA and P2A; (c) The preliminary sta- 
 tion and plus for A is PiP 2 + P^A; and the loca* 
 tion station and plus for A is PiA. 
 
 8. Having drawn on the map the check line 
 PCsP?, compute the length and bearing 'of this 
 line. This is done by treating PCaP? as the "miss- 
 ing side" of the closed figure APaP^PriPftPTPCs- 
 
 VaViA. 
 
 9.' Write the alinement notes in the field book. 
 
 The work of finding the "missing side" in Step 8, 
 just above, involves practically all the operations 
 required to compute the co-ordinates of the chief 
 points of our map from the field notes of the pre- 
 liminary survey. 
 
 Computing from the field notes the rectangular 
 co-ordinates of the chief points of a survey is 
 analogous to computing from the level notes the 
 elevations of the stations on a profile, and has 
 similar advantages. 
 
 18 
 
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