64 Mr HOLDITCH, ON ROLLING CURVES. 



If r - - = x, any component part of X as 



s 



=(i-r"(i-n(i-)" + (i-n 

 =s--r-{(i)"-- i i i -(r-^ }• 



consists of even powers of x only, and therefore X will contain no 

 negative powers of x, and will be of the form 



a + b\ 2 



X(r, c -r) = k l + k(r- ° L ^-) + 



and limiting the investigation, for the sake of simplicity, to the first 

 two terms, 



k t + k.{r- t±Jj 



we have dO = . . dr. 



ry/{a — r) . (r — o) 



To integrate this, let — - — m a, — - — = /3 ; 



•. (a - r) . (r - b) = (a + b) . r - r 2 - ab = 2ar - r* - a 4 + /3 s = p - (r - a) 2 ; 



/V/3* _ (/• - a) s 



, A + £)3*cos*d> , , 

 Assume r — a = /3 cos d>, then a0 = ' ^ :r^ • a< P 



r r a + p COS 



£, + £ . (a + /3 COS — a) 2 -. 

 a + y3 COS (p ' ™ 



= '—p: . d<t> + kadcb — kfi cos <bd<b. 



a + ft COS (p T T T T 



