1882.] On Hypocycloids, Epicycloids, and Polyliedra. 107 



Journ.," vol. iii, p. 145) is liere adopted. Let x 2 + y 2 =p, 2xy = a. 

 Cotes' theorem is thereby applicable to determine the product. 



x~ H + y 2n — 2x?y n cos nO=(x%— 2xij cos + 1/) | x- — 2xy cos^— + • • • 



"We must next express x 2,l + y 2 ' 1 in terms of p, a. The expression, 

 which is closely allied to the expansion of cos n0, is given in Tod- 

 hunter's " Theory of Equations," p. 183. 



Since (1 - zx*) (l-zif)=l-z (a* + *f - zxhj*) , 



if we take the logarithm of both sides, and select the coefficient of z n r 



It is also necessary to obtain the ascending series — 



,^ + «=d)--^)-«@"+fssGr(s)'- • • • 



if n be an even integer, 



^^=-S)">¥t¥6r© ,+ • • • 



if n be an odd integer. 



7. The polar tangential equations to the regular hypocycloids and 

 epicycloids are 



n + 2 . n 

 p= — — a sin 



n+ 



where a is the radius of the fixed circle, and bears to the radius of 

 the rolling circle the ratio n : 1 (Besant, " On Roulettes,' ' § 14). 

 If we take the upper sign, and write — 



sin 



Hence — 2 cos nO 



■ z cos 



= (2sin x )"" 3 -0t-2)(2siiix) M " 4 + ^ f ^ 5) (2sin x )»-6-. . . .. 



\(n-2)a) K J \(n-2)a) 



Hence the general theorem is established for hypocycloids. 



