Mb HOLDITCH, ON ROLLING CURVES. 63 



Also, if c be the distance of the centres, r + r, = c, and 



... dr* + r*d<? = df (l + ^) = drj (l + ^*) - drf + r •*,•. 



or the differentials of the lengths of those parts of the curves which 

 have been in contact are equal, and 



, rde rde, 

 .-. r + r= c, and -*— = -3 — - , 



are equations which contain the analytical conditions of such curves. 



We will first consider the case of two equal and similar curves 



rolling on each other. Since -j- is some function of r, *%-* must also 



be a function of r, let it = fir) ; and as r, and 0, belong to a point in 

 a similar and equal curve, 



••• -^r =/CO; and r, = c - r; .: f(r) =f(c - r), 



the solution of which equation is f(r) = <j> (r, c — r); any symmetric 

 function of r and c — r, and if any form be given to <j> in the 



equation —7— = <p(r, c - r), the integration of the latter will give the 



equation to a curve having the proposed property. If we suppose it 

 to have greater and less apsidal distances a and b, which most curves 

 which can practically be used, must possess ; then, as in revolving the 

 greater apsidal distance of one must come into contact with the less 

 apsidal distance of the other, a + b = c ; 



Now (a - r) . (r - b) = (a + b) . r - r* — ab = r . (c - r) - ab, is a 



dr 

 dd 



symmetric function of r and c — r ; and as at apses -73 = 0, if we 



rd6 X(r, c — r) , __. N . . . 



assume -7— = , v y , where X(r, c - r) is any symmetric 



dr V(a — r).{r — b) 



function of r and c — r which is not divisible by \/(a — r) . (r — b), 

 the curve will be confined between the apsidal distances; and sup- 

 posing also, that X contains only positive integral powers of r and 

 c — r, this last equation can always be integrated in finite terms. 



