76 Mk holditch, on rolling curves. 



1111 



R at + R b , a + f; 

 R a / R b a, + b> 



JL _L I I 



R a + R hl - a + b, ■ 



. , 1 s , ds 



Also, as y» - +tt» 



,1 s ds s ds # 



'*• R r + Rr, ~ r + r, ' 



Ify(^) De the radius of curvature of a curve at a tangent point; 

 the radii of curvature when r becomes r + h, are 



R=f(r + h)=f(r)+f'(r).h, 



and, the corresponding radii of curvature of another curve rolling with 

 the former, are 



R, = <i>(r i ±h) = <t>{r)±V{r).h 



= <p{r) + <p x (c — r).h; 



also, since the radii of curvature of the two curves at the tangent points 

 are equal 



/(r)«*<r,); 



and therefore R - R t = + \f x {r) - <f> l {c - r)}.h. 



Hence, if R > R t before the tangent points come into contact, 

 R < R, afterwards, and consequently the curves cross and change their 

 rolling sides at the tangent points: except h t = 0, when there is a point 

 of contrary flexure at the tangent points, which then also coincide at 

 the mean distance. 



