60 C. J. MERFIELD. 
To find the exact radius of curvature in terms of « and y. 
ie 3 +9 m? nett 
means 4 
6m2z 
substitute for m its value —**, “3 and we obtain 
sotaye V (ae? +9 ye?)® 
p= R= JES x“) = < fae. ieeeks 19 
ee ci 
To obtain p at any point on the curve, it is only necessary to 
place the values of the co-ordinates x and y in equation 19. 
TABLES. 
Table 1 contains the data necessary for the location of the 
transition by the exact method. The first column gives the dis- 
tance x’ = EF B,EO =x —EB. H equals the distance TL 
of the parallel tangent, this quantity is to be set inwards, after 
the lines O LZ, L O', have been fixed. This latter distance 18 
obtained in the following manner 
LH=LE' = tan © (R+H) 
LO=L0O = tan > (R+H)+£O (See fig. 2.) 
The angle ¢ is also tabulated. 
The offsets at various distances then follow. OC equals length 
of transition. equals twice the circular measure of angle ¢, | 
this quantity is taken from the circular measure of the angle % 
the result is the circular measure of (o — 2¢). This quantity 
being multiplied by the radius, gives the length CC’, All dis 
tances are in links, except where mentioned. 
Table la and 16. These tables are constructed in the same 
manner as above, and will be found useful to the tramway engineer. 
Comparing these two tables with Table 1, it will be found easy 
to extend, as may be required. 
Table 2 is similar to Table 1, but by taking « = 2, we 
eliminate the first quantity given in Table 1. The angle ¢ is not 
tabulated, but may be obtained from the equation tan ¢ = sa 
