Mr HOLDITCH, ON ROLLING CURVES. 71 



it follows from what has been observed before, that 



r.d 9 ' . 



will be the differential equation of the latter; and any form being given 

 to f, the integration of these equations will be the equations to a pair 

 of rolling curves; and for other values of c, other curves may be found, 

 and so a system formed. 



The equation to one of the curves being assumed to be that which 

 has been found for self-rolling curves, viz. 



It + k [r 



rdd ' \ 2 



dr ' y/( a - r ) . (r - b) 

 the equation to the other will therefore be 



r'de. *, + *-(«-',-nr) 



dr, " y/( a - c + r[) . (c - r t - b) ' 



let c — b = a,, and c — a = b\ ; .*. a,— b,= a — b, 

 , a + b a + b, 



. r M ^'-(H 3 ---)' '.+ '•(-■-'4^' 

 dr, n/(u t -. r ).{r t -b) Via,- r t ).(r,-b t ) 



which is of the same form as the differential equation of the assumed 

 curve, and therefore if n, be the number of its lobes, 



,,-g +*.(..+*,)}. <»- V|.VJ£* 



- *V(a,- r).(r,-b)-k.(a l +b).tzrr\ V^rf. 



a, - r. 



