1846.] 



THE CIVIL ENGINEER AND ARCHITECrS JOURNAL. 



275 



distant from each other as he passed over a common which lay in his way ; 

 not knowing the comparative length of his proposed line, he was not coo- 

 fined to any particular distances. Now it so happens, that he sets up a 

 staff at each of the four points C, D, E, F, as in the annexed sketch, and 

 continued the measurement of the line. A second assistant who measured 

 a tie line from G to K, before leaving G, took the angular distance be- 

 tween E and F at IG" IG' 16"; this was the only angle he could take from 

 the point G, he not being able to gee C or D, though from the direction Clay, 

 he knew the angle F G C, to be acute ;— then measuring from G to K, his 

 line exactly passes through the station D ; hut when I met him at H, the 

 chain he was using was not correct. Therefore, the only particular we 

 knew of this line was, that G D happened to be the same number of false 

 chains as D H. Leaving this assistant to take the angle E H D, which he 

 makes 34° 34' 34', I measured H K, with a correct chain and found it to 

 be 1500 links, and at the station K, I took the angle F K C at 70" 10' 10" ; 

 not being able to see T> or E. It is required from this data to find the 

 length of the different lines; not having any information from the first 

 assistant as to the distance he placed the pickets, and ou returning to re- 

 measure the distances, the exact place where they stood could not be as- 

 certained. 



Now, we have given, the distance CD = DE = EF ; and DG = DH ; 

 also HK = 1.500 links. Likewise, Z FGE = 16° 16' 10" (a) ; EHD = 

 DGC=31°34'34"(i); and FKC = 70° 10' 10" (c). Let Z DGE = 

 X, DEG = y, CKG = s, and ERF = v. . • . CDG = {x + y), FKD 

 = (c — :), and EKG = (c — ti). Hence, by comparing the ratios round 

 two points C, and E, we have 



CD : CE = 1 :2; 

 C E : CG = sin. (4 + ■^) = sin. y ; 

 CG : CD = sin. (x + i/) : sin. l ; 

 FE : FD = 1 : 2 ; 



FD : FG = sin. (o + ^) : sin. (x + y); 

 FG : FE = sin. y : sin. «. By com- 

 pounding these six ratios and expunging 

 the common factors we have, sin. (a + x) 

 sin. (6 + jr) = 4 sin. a sin. b. cos, (i — a) 



— COS. {(a -f i) + 2.r} ^ 2 sin. (a + a') 

 sin. {b -\-x) = 8 sin. a sin. 6. Cos. 

 (a — b) — 8 sin. a sin. b = cos. 



{(a + 6) + 2.r}. But, — 8 sin « sin. 



i = 4 cos. (a — i) + 4 cos. (u + ii) ; . • . 4 cos. (a + i) 3 cos. (a — b) = cos. 



{{a + b) + 2x} . An arc whose cos. is 4 cos. (n+b) = 3 cos. (a — b), 



written thus; — Cos. 4 cos. (a-hi) = 3 cos. (a— i) ; made less by 



(a+b) =2x. .•• 2.r = cos. ^{4 cos. (a -f- 6) — 3 cos. (a — b)} —a—b. 



Whence tiie Rule.— From /o«r times the nntural cosine of (a + b), take 

 three tintes the natural cosine of (b-a) ; an arc whose natural cosine is the 

 remainder, made less by (a + b), and divided by 2, will gire x. 

 (a+b) = 50° 50' 50", 4 nat. cos. = 2-5235C16"l 

 (4-a) = 18° IS' 18", 3 nat. cos. = 2-8481913/ 



subtract. 



Difference = 0-322G327 negative. 

 The angle which corresponds to this nat. cos. in the table is 71° 10' 41'' ; 

 but as the cos. is negative, the angle may be 108° 49' 19", or 251° 10' 41"; 

 bnt it cannot be the latter, because the angle FGC is known to be acute. 

 Then from 108° 49' 19" 



take 50 50 50 = (a -f i) 



half of 57 58 29 = 28° 59' 14'-5 =:ar= KGE. 

 In order to determine y, let us compound the ratios of the lines drawn 

 from E, which are compared in the foregoing ratios. Hence we have sin. 

 (a -\- x) sin. y = 2 sin. (.«■ -|- y) sin. a. But, sin. a cos. x sin. ;/ -f cos. a 

 sm. X sin. j/ ^ 2 sin. x cos. y sin, a + 2 cos. x sin. y sin. a. This divided 

 by sin. y, gives sin. n cos. x -f- cos. a sin. .r = 2 sin. x cot. y sin. a -j- 2 

 cos. X sin. a. . • . sin. a cos. x 4- cos. a sin. x — cos. x sin. a = 2 sin. x 

 sin. a cot y ; sin. x cos. a — cos. x sin. a = 2 sin. x sin. a cot. y. .' . 



sin. (x—a) 

 it sm. a sin. x ~ '^°'" ^' '^''°'" ^'^ '"'^'"^ "*" '^°'' ''' "^^ following method 

 of calculating y is deduced : Rci.e. — Add together the log. sine of (x—a), 

 the sub. log, of 2, the log. cosecant of x, and the log. cosecant of a ; the sum 

 will be the log. cotangent of y. By this rule y is easily found, and x is I 



known, — call their sum m. [Which will be found too by 79° 57 35"-5]. 

 Then to find z, we have 



FD:FC = 2:3; 



FC; FK = sin.csin.(OT-2); 

 and FK ; FD = sin. m sin. (c—z) ; 

 .-.2 sin. csinm = 3 sin. (m— :)sin. (e — z; 

 cos. (m-<:)-cos. {((n + c)-2i} = 2 sin. (m + :) sin. (c-t). 



^ 2 S'°. « sin. m; 

 cos. {m—c) — \ sin. c sin. jn = cos. {(,« — c) —2:}. 

 But, COS. (in-i;)-cos. (m -j- c) = 2 sin. c sin. m 

 . • . f cos. (»n-c) — f COS. (m-\-c) = ^ sin. c sin. m 



J COS. (m-(;) + f COS. (m-fc) = cos. {{m^c)-2z}. 



.-. 2z — (m + c)— COS.— i {cos. (m—c) -^2 cos. (m + c)}, 

 EcLE. — Add together the natural cosine of (m — c) and twice the natural 

 cosine <if (in + c), and divide the sum by 3 ; the result will be the natural 

 cosiJie of an angle, which angle taken from (m + c) will leave twice z. 



This gives the angle = 22» 50' 14"= » = CKG. To find the angle 

 GKE = v, we have FKG = (c-z) = 47° 19'56" = n ; FDK = (x-y) 

 = 79° 57' 35"-5 = m. 



We have also, FE : FD = 1 : 2 ; 



FD : FK = sin. n : sin. ra; 

 and FK : FE = sin. (m + v) : sin. (n-t). 

 .• . sin. n sin. (ra + f) = 2 sin. m sin. (n — v); 

 sin. n (sin. m cos. i; + cos. m sin. d) = 2 sin. m (sin. n cos. »— cos. n 

 (sin. v). Sin. n sin. n cos. v+ sin. n cos. m sin. i') = 2 sin. m sin. n cos. 

 f— 2sin. m COS. n sin. ». Dividing by sin. n sin. m sin. r, we have cot. » 

 -fcot. m = 2 cot. »— 2 cot. n; 



. • . cot. m -I- 2 cot. n = cot. r : which affords the Rule. — To the natural 

 cotangent of m, add twice the natural cotangent of n ; the sum ivill be the 

 natural cotangent of v. . • . ti = 26° 19' 55." All the other angles of the 

 figure can be found by addition and subtraction; and any of the linear 

 distances calculated at pleasure. EH, ED, and GK, will be 464013 

 26-7423, and 100-7355 chains, respectively. 



THE PRACTICE OF SETTING OUT RAILWAYS AS PRE- 

 LIMINARY TO THE CONTRACT WORKS. 

 The duty of an engineer when setting out a line of railway, finally 

 and permanently, has not, so far as I am aware, been described in any 

 publication. It may be serviceable to some of the readers of the Journal 

 to show in what it consists, and to impress the evident fact that its careful 

 and accurate performance is important in every department of the construc- 

 tion of the line, and to the interest of the railway company in relation to 

 the landowners. Can anything be more discreditable than a distorted or 

 irregular alignement ; — Uian a deviation from the gradients established on 

 the permanent seciion, or than winding slopes ? Can anything be more 

 injurious to the character for honesty, which a railway company, ihrou^h 

 its servants, ought to maintain, than the frequent squabbles with land- 

 owners as to tlie quantity of land purchased from them? Discreditable 

 however, as these exceptions to propriety are, instances of them all are 

 familiar to the practical engineer. They are in many cases attributed 

 perhaps to the hurry of commencing the contract works most fre- 



quently they can be traced to the want of experience or of care on the 

 part of those entrusted with the duty of permanently setting out the line. 

 To assist the inexperienced and to direct attention to the points which 

 chiefly require care, is my purpose in oli'ering a contribution to the pages 

 of the Civil Engineer and Architect's Journal, upon this part of an en- 

 gineer's practice, and it has been my object to avoid all, but the simplest 

 forms of calculation, to reduce them to the fewest in number, and to show 

 how they may be made with the least trouble — in fact, so as to save all 

 the trouble that it is possible, consistently with accuracy. 



A line of railway is permanently set out,— First, by tracing upon the 

 ground the centre line, which must conform to the final plan, as to curves 

 and tangents. 



Secondly, by planting permanent bench marks to indicate the gradients 

 laid down on the final section. 



Thirdly, by marking off the widths so as to show the space occupied by 

 the railway at formation level, or balance line, by the slopes, and by fence 

 and ditch, from the dimensions presented by the cross sections. 

 To the two last operations it is jiroposed to confine this paper. 



