342 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Calculating, and substituting the values of l , m, and n, the equation of the 
transition curve becomes 
y = -000, 000, 31294a 3 - -000,000,000,01145a 4 4- *000, 000, 000, 000, 0292a 5 . 
The curve is then set out in the ordinary way. 
A Method by use of Intrinsic and Freedom Equations 
to the Transition Curve. 
If the curve chosen be such that the curvature changes uniformly in 
passing along the curve, then 
rf /h=. 
ds\p. 
So with a properly chosen origin 
dlfr 
ds 
= cs, and if/ = Jcs 2 , or s = 2ai^-, 
the constants of integration being zero. This is the intrinsic equation of 
this transition curve. 
(jfs 
Since p = , at the point of compounding a = R^/q% which determines 
the constant a. Also we have 
dx = cos if/ ds = ^1 - 1 - . . ^aif/~*dif/ 
dy = sin if/ds = ^ + • • ^aif/^difr . 
Integrating, and remembering that at the origin 
® = 0, y = 0, if/ = 0, 
the freedom equations of the curve are found to be 
*-“( w-h.-b t+ h-b * — ) 
This does not allow of exact compounding both at an arbitrarily chosen 
origin and at the P.C.C. When the P.C.C. is chosen, the origin and shift 
are thereby fixed. In other words, the curve represented by \]s = Jcs 2 does 
not allow of compounding in the most general case. 
It may be used, however, to give exact compounding when the P.C.C. is 
chosen arbitrarily and the origin deduced. This method possesses no 
advantage over the method of exact compounding by the cubical parabola, 
and the calculations required are more laborious. 
