1882.] On Hypocycloids, Epicycloids, and Polyhedra. 109 J 



x I x x* x 6 J 



1W x- I lx 1.2 »s " • " J 



T 1.2W « 5 1 laj 1 . 2 su3 " ' " J ' ' ' 

 By equating the several powers of -, the several sums of powers are- 



found. 

 2 



This is the dual of Stewart's Prop. 40. 



13. If in § 12 the line touch the circle, so that p = a, 2(p,) m be- 



(2m— 1) (2m— 3) . . . 3 . 1 m rT > n QQ x 



comes n- — — J - a m . . (Prop. 39). 



m(m — 1) ...2.1 



Lemma. — Every product of consecutive factors can be expressed as a 

 sum of a product of lower consecutive factors. 



Thus l + 4( lt -4) + 6 (M ~ 4 > fa ^ +4 ^~ 4 >^- 5 ^"- j) 

 1.2 1.2.3 



(n-4>)(n-5)(n-6)(n-7) _ n(n-l)(n-2)(n-3) 

 1.2.3.4 1.2.3.4 



as appears by the equating coefficients of x 4 in the identity 



(l+aO"- 4 (a + l) 4 = (l + aO w . 



Hence when p = a in § 12, 



(1 + 1 \m + om(m-l) a 1 x m _ 2 , 4^3 m(m-l)(m-2) (m-3) a -, . m _ 4 

 ' 1.2. / 1.2 1.2.3.4 ^ ^ ' 



+ . . . 



= l + TO{ l + (TO -l )}+ !^!^|l + 2(m-2) + ^^ . . . 



= i 2 + (*Y'+ J ^-l) l 3 , - 1 .3...(2m-l) 

 VI/ I 1.2 J * 1.2.. . m 



14. Dr. Stewart's general theorem (Prob. 42) follows from the same 

 formula as the former. 



II. On Theorems relating to the Regular Polyhedra, which are analogous 

 to those of Br. Matthew Stewart on the Regular Polygons. 



1. These two general propositions may be thus stated in the dual 

 form : — 



