Mr HOLDITCH, ON ROLLING CURVES. 73 



In this latter example, it will be observed, that the curves are re- 

 trograde at the apses, which will be the case with all unequal curves 

 that are made to roll together, if they have the same number of lobes ; 



or ' + ' vi/ r y^,.(v^ - \Zb,y - s ~a~b. (Va - \Zby 



from equations (2), and if n = « , this expression may be proved to be 

 negative when a and #,, and therefore b and b /} are unequal: and since 



and ~*HVa,-^,r=^.{k, + k.(H^)); 



therefore, by equations (4), the curves are retrograde at the apses. 



If two curves roll one within the other round fixed centres whose 

 distance is c, then 



h h ( a + b y 



. rd9 r,de t , .„ rd9 «< + *'\ r - 2 ) 



r = r + c, and — t— = -~j — * , and it —>— = — , , _ — , 



' - ' rfr dr, dr y/{a-r).(r-b) 



be taken for the differential equation of one ; 



a + b\ 



rde t 



iil a + by 



& t + k.[r, ± c —J 



dr, *S( a -r / + c).{r l ±c-b)' 

 will be that of the other; 



let a + c = a, ; and b + c = b, ; 



, a + b _ «, + b, 



... « _ J) = a 7 - 6 , and — — + c = — ^— ; 



and'-*. 



*,**-{4M ^ »•.('. -^ 



<*r, </(«, _ r) . (r, - b) V{a, - r,) . (r, - b) ' 

 Vot. VII. Paet I. K 



