8 CURVE OF 



. myp 



Then by substitution, -r^ ; — zza, 



V (n z^ + mp' ) ^(?ix — mp) 



Cor. 1. Whenm=o; this expression becomes 

 ~— •=. a, the equation to a cycloid. 



Cor. 2. If in the equation 



• .myp 



a, n — % and m = a = 1 ; then 2 p y 4- y p = 



p >/(2 a: — ;?) . V{2 z^ 4- p2;) and squaring both sides, 

 4p2y2 4-4pyp^-|-i/2p2--(4p23._2p3)z2 4. (2 p- x — p^) p^. 



Buta;2=p2 — y2. and taking the fluxions, xx = 

 V V — y h ^^^ squaring both sides x'^ x^ = 

 pp^ — ^pyp'y + y"^ y^ : and therefore 4 py py = 

 2p2 p2 + 2y2 y2 _ 2 a;2 a;^. And by substitution, 

 (6 p2- 2 x2)y2_ 2x2^2— (4p2a:~2p3) z2 + (2p2a;— 

 p3— 3p2-f a:2)p2.(l). 



But «2 _Z£Zl£f r _ p2p2.^2pa:pa: + a;2a;2: 



