INTRINSIC EQUATION OF A CURVE, AND ITS APPLICATION. 671 



2 4 6 



a a a 



= 1--—+ + &c. 



1.2 1 ... 4 1 ... 6 



whence 



a" . a' 



Hence the necessary relation among 6,, 62, b^, &c. is satisfied if 



b„ = m'b„_, = TO"6„_2 = m''b„^, = m"b„.„ 



that is, if A..,= — fi„, 6„-o = — 6„, 6,-3 = — 6„, &c. 

 Hence we have, by the expression for s„, ultimately, 



.s„ = mb„ |2 i— 2. + _J_ 2. _ gjc.l 



Ito 1 .2. 3 ot^ 1 ... 5 m" , j 



• 



= »»6„ sin — . 

 m 



This is the equation to an epicycloid, if m > 1 ; and to an hypocycloid, if wj < 1. 

 If A and B be the radius of the globe and wheel of the epicycloid ; 



iB{A + B) . A ^ 



s = sin — =r (p. 



A A + 2B^ 



„ A + 2B B m-\ 



Hence m = ; — = . 



A ' A 2 



r. u u , J iB{A- B) . A 



ror the hypocycloid, s = sin <B. 



^^ ^ A A -2B^ 



A - ZB B 1 - m 



Hence m = : ; — - = ■ 



A A 2 



In the figures, the angle ACB = (m - 1) a, when m > 1 : 



ACB = (1 - m) a, when m < 1. 



t b2 



