58 _ C. J. MERFIELD. 
If the circular measure of the angle ¢ is now taken, and — 
multiplied by the radius of the circular arc, we obtain the length — 
of are 7’ C or 7’ C’. 
Now length of are 7’ 7" equals the circular measure of the angle — 
w multiplied by the radius. Hence the following formula 
CC'= R§[o—2 (Sin —1 £)]}-0002909 12 
The angle within brackets being expressed in minutes. 
Approximate Method of Location. 
We shall now proceed with the investigation of an approximate — 
method 
2 
pas = ga: 1 
6 me 
1 
Therefore x = eI ead OF 
In the preparation of the tables (excepting table la and 16) # 
has been taken equal to four chains, this being the value adopted — 
by the Railway Construction Department. But really « could 
be varied inversely as the radius. 
That is to say, if we use four 
chains as the length of transition to the ten chain radius arc, W — 
would obtain the same result, by taking two chains as the length ‘ 
of easement to a twenty chain radius are, providing the velocity — 
is the same. However it is perhaps as well to keep « a constant 
value, and so avoid complexity. 
To locate the position of the parabola. Let the circular are ¢ C 
be produced to 7, where its tangent becomes parallel to the axis 
of X. Then the angle 
CDBe Crt => 
Now tan > = aes == au om 3 7 Be" 
¥o = | 5" 7% = Be 
Therefore tan ¢ = ee 6 
The tangent at the eh C, if ~— to the axis of X, cuts 
it at D making O D = 3,, and D B= ia, (See fig. 1) 
AgainCG=EFB=(R- GT) tan ¢ 
Approximately H B= FR tan 4, taking G 7’ as a small quantity 
