396 MESSES. C. AND E. CHAMBEES ON THE MATHEMATICAL EXPEESSION 
to which adding cos 0/3, we have 
0=cos 0+cos/3+cos 2/3+cos 3/3+ +cos(w— 1) (3,1 
according as /3 is not or is 0 or a multiple of lir ; 
or (say) SX=Y, 
S=YX" 1 , 
dS 
^-=sin (a + /3)+2# sin (a + 2/3) + 3# 2 sin (a + 3/3) + 
+ (w— l)#" _2 sin{a+(w,— l)/3f 
-fx--ygx-, j 
which, when x—\ and a= 0 (see equations (h), (i), (1), and (m), 
=2 -1 (1 — cos /3) _1 {sin /3— wsin w/3+(w+l) sin (n— l)/3[ 
— 2 -2 (l — cos /3) -2 [2(1 — cos /3) -{ sin /3— sin w/3+sin(w— l)/3 }] 
=2 -1 (1 — cos /3) _1 [— (n— 1) sin w/3+w sin (n— 1)/3], 
which, when n(3=2cT, c being an integer, 
n sin B n B 
= ~~~r^ = ~2 cot 2 I 
4 sin 2 ^ 
= sin /3+2 sin 2/3 + 3 sin 3/3+ +(n— 1) sin (n— 1) (3 ; 
and as 0 sin 0/3 = 0, 
% m sinm/3= —xcot^, when /3 is not 0 or a multiple of 2x. . . 
But when /3 is 0 or a multiple of 27 t, each term of the series is 0, and 
% msinm/3=0 
Now let x=l and a=~, and (6) becomes 
^=2 _1 (1 cos /3) _1 -{ — cos /3— n cosm/3+(m + 1) cos (n— 1)(3} 
— 2~ 2 (1 — cos|0) _2 [2(l — cos0)| — cos j3— cos w$ + cos(w— l)0 + l}]j 
(4) 
(5) 
( 6 ) 
(7) 
( 8 ) 
(9) 
(9«) 
m 
( 10 ) 
= 2 ‘(1 — cos/3) 1 [— (n— 1) cosw0+w cos(n— 1 )jQ — 1], 
which, when w|3=2c7r, 
= 2 _1 (1 — cos0) 1 [-(n—V)-\-n cos0— 1]=— - 
= cos /3+2 cos 2/3 + 3 cos 3/3+ +(w— 1) cos (n— 1)/3 ; 
( 11 ) 
