334 Proceedings of the Royal Society of Edinburgh. [Sess. 
so that a 0 , a v and a 2 are all zero. Hence, a solution may be obtained in 
the form y = 'Ea 3+ir x 3+Ar . This shows that a very close approximation near 
the origin is given by the cubical parabola y = lx 3 , the subsequent terms 
in the expansion being of the nature of correction terms. It is essential, 
however, to get exact compounding at the termination of the arc. It will 
be shown that it is possible to obtain this required exact compounding 
by using the cubical parabola in a particular way as a transition curve, 
Fig. 1. 
and yet under conditions which are, from the engineer’s point of view, 
almost perfectly general. A further method of complete generality will 
be considered later. 
Notation. 
ABLX, original tangent or line of straight track, taken as £-axis or 
initial line. 
FDM, parallel tangent, from which the actual circular curve is set out. 
BD = AF = BT)', the shift as calculated (/3). 
AEQ, transition curve (T.C.). 
