Mr HOLDITCH, ON ROLLING CURVES. 81 



The distances may also be found practically, by describing an ellipse 

 whose axis major is «„ + b n , and «„ — b n the distance between its foci ; 

 then if straight lines be drawn from one of the foci to the ellipse making 

 equal angles with each other, and the base of the lobe be divided 

 into as many equal parts as there are equal angles round the focus : 

 the distances from the centre to the several points of the lobe are 

 easily shewn to be equal to the elliptic distances ; and may therefore 

 be set off from them. 



The form of a rack, or curve of an infinite number of lobes to 

 move with the curves derived from the equation 



b 

 r 



Wab T 2\/ab I b v a- 



- k.y/{a — r).{r - b) - k . (a + b) . tan" 1 \/Lz3L , 



a — r 



may be found by making n infinite and a — b = I, where a and b are 

 also infinite; and this form is that to which the lobes gradually ap- 

 proach as n increases : if x and y be rectangular co-ordinates of the 

 rack, x being measured along its base from one of the apses, and 

 y be perpendicular to the base, x = bO and y = r — b\ 



= (a*, \/l + Wl . (a I by) . tan- \/\. \fX 

 [ ' a 2 a ' ) b I — 



x 



- bks/ly - f - k.{ab + ¥) . tan" 1 Vr^— . 



By Maclaurin's Theorem, the expansion of tan -1 V x- 'V r~^ — » 

 b l — y 



of tan -, (l + t) • V i _ as far as the square of t is 



-^•^-("♦•"♦'S)-(i-£*S) 



Vol. VII. Part I. L 



or 



