DR MATTHEW STEWART'S GENERAL THEOREMS. 581 



In the first lemma, by making r v =r, we should get 



(r 2 - 2 r /j cos w + rf) n = (2 r 2 )* (1 - cos )" ; 

 and in the second, by making p=r, we should get 



(p r cos w)" =r"(l cos w)". 



We should thus obtain two different forms: but the expansion will be better 

 adapted to our present object when obtained as follows : 



Since(l cosw)"=2"sin 2n - w ; the ordinary formula, making the requisite 

 change in the form of the last term, gives 



siu 3 " - w = ( -_ 1 - )n22K - 1 ' { cos 2 n - w - -f cos (2 n-2) ^ w + ---- 



' 2' 1.2.3 

 In 



, . 



{ > ' ' 



2 ' 1.2.3 ..... n 

 (for 2 n is always even, when n is an integer.) 



If 2 1 



= (-l)"2 a "- 1 t COSMW ~ -j- cos (-!)&)+ ---- j 



J_ 2(2n-l) ---- 

 2 2n ' 1.2.3 



2w 



If 



= (-l)2 2 "- 1 \ COSM 



l^ 1.3.5 ____ (2-l) 

 2 '1.2.3 ..... n 



LEMMA IV. 

 If ^i ^2> ^ 3 , .... 5, be n angles in arithmetical progression, whose common dif- 



2/Tj- 



ference is (or, which is the same thing, if they be the angles formed by lines 



from the summits of a regular polygon with any arbitrary line, the centre of the 

 polygon being the origin of co-ordinates), we shall have simultaneously, 



COS <9j + COS $2 + COS 3 +.... + COS 6 n =0 



cos 2 X + cos 2 2 + cos 2 6 3 + ---- + cos 2 6 n =0 

 cos 3 6 l + cos 3 6 2 + cos 3 3 + . . . . + cos 3 & =0 



cos (-l) 6 1 + cos (-l) 2 + ---- + cos (w-1) & = 

 VOL XV. PART IV. 7 s 



