340 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Table of Distances in Feet for Selected Vectorial Angles. 
0 
2' 
7 
15' 
30' 
40' 
50' 
1° 
r 
54*572 
102*10 
149*45 
211*37 
244*08 
272*90 
298*97 
Again, suppose that instead of g the approximate value of the shift is 
given, and instead of the degree of curve the value of R. 
Example . — Given R = 1000, <0 = 58° 50', and 8 = 1*35 (approximately). 
Then 
and hence 
Take 
if/ = 5° 9' 24". 
if/ = 5° 10' 
for ease of calculation, and redetermine 8- 
Then 
£= R sm\J/ = 90*0532. 
r, = R(l- cos if/) = 4*0630. 
x = 2R sini^cos 2 if/= 178 ’646. 
y = §R sin 2 ^ = 5*38442. 
1= l/{ 1 2R 2 sin if/ cos 5 if/} = *000,000, 944, 41 . 
Hence 
P = V - >7 = 1*3214, 
a = x- £=88*5925, 
and the equation of the curve is 
Y = *000, 000, 944, 41X 3 . 
As SB = (R + /3)cot o> = 605*626', the P.C.C. is determined and the points 
A, N, Q, D are all fixed. Hence the circular and transition curves may be 
set out in the usual way. 
Table of Offsets. 
X 
20 
40 
60 
80 
100 
120 
140 
150 
160 
170 
180 
190 
200 
Y 
*008 
*060 
*204 
*484 
*944 
1*63 
2*59 
3*19 
3*87 
4*64 
5*51 
6*48 
7*56 
Exact Compounding. P.T.C. and P.C.C. both fixed. 
As the cubical parabola cannot be made to satisfy these conditions, an 
equation containing more arbitrary constants is required. Referring to the 
original differential equation, and considering terms of higher degree in x 
