Mr HOLDITCH, ON ROLLING CURVES. 75 



and d ^=2k.{r- ~^j =2k.\/^ when X = 0; 



also Y — (a — r) . (r — b) = — — [r —J = — + -^ where 1= a — b; 



therefore, if R = radius of curvature at the tangent points, 



1 dp ds 2k y/ — k t 



+ k 



R rdr dr / /«' 



V *,■ 



4 



which depends only on a — b, and therefore, the radius of curvature is 

 the same at all tangent points of curves of the same system. 



At apses, Y = 0, and s = 1 ; 

 2ds 1 dY 



s>dr X*dr' 

 l_ 



as i , , _ . i .„ 



d> = -2X-*' {a + b - 2r) = „f, ,JV ' lf r = fl ' 



and = 5 7^ ' if r = *• 



Let R a represent the radius of curvature, when r = a ; 



«?.? 1 



R. r dr a 



2 



(*, + *i) 



Tf ~ h I / 2 \ 2 * ' )" 



Hence also the following equations: 



1111 

 /J,, + R b == « + b ' 



