338 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Thus to a first approximation x x — 2£ or AB = BN approximately, and 
this is nearly half the length of the transition curve. Thus when the 
length of the transition curve is given by practical considerations, the 
positions of the P.C.C. and the P.C. are both fixed within very narrow 
limits. 
Also as 
7] = R( 1 — cos i f/) ; y — § R sin 2 i f/ cos if/, 
expanding in ascending powers of \fr. 
rj = R 
r_ t , 
2 24 
and hence to a first approximation /3 = -JRi/A Thus when two of these 
quantities are known, the third may be found, as when R and \js are given 
the approximate value of the shift is obtained at once. 
The simplicity of the preliminary trial calculations and the wide range 
of selection combine to render this method one of extreme flexibility. 
An approximate value of the length of the curve to any degree of 
accuracy is easily obtained. Thus to a second approximation 
s = x+ '9y 2 /x. 
Example. — 2° curve, angle between tangents 94° 38'. Compounding to 
occur near ^=150'. 
Then 
R = 100/-0349066 = 2864-787'. 
sin ^ = 4 = 150/2864-79; .*. ^ = 3° O' 5". 
R 
As /3 = JR \fs 2 approximately, we have 8=13' nearly. 
For simplicity of calculation take \]s = 3° as the value which determines 
Q, the point of compounding. Re-calculate £ ; hence 
£=149-932, ^ = 3-926. 
For the co-ordinates of the point of compounding 
Hence 
x = 2R sin ifr cos 2 ifr, y — |R sin 2 if/ cos if/. 
* = 299*041, ?/ = 5-2240. 
For the position of the origin, 
a = x- £= 149*109, fi = y - 77 = 1-298. 
For the equation of the curve, 
1 2R 2 cos if/ sin 5 if/ 
