398 
MESSRS. C. AND E. CHAMBERS ON THE MATHEMATICAL EXPRESSION 
But when (3 is 0 or a multiple of 27 t, each term of the series is 0, and 
% m 2 sinm/3=0 (176) 
771 = 0 
Now let #=1 and a=^, and (14) becomes 
dxif^) (l — cos/3) -1 [cos/3— % 2 cosw/3+(w+l) 2 cos (n— 1) /3— 4] 
— 2 -2 (1 —cos /3) -2 [{ 3 cos (3— (2w+l) cos w/3+(2w+ 3) cos [n— 1) /3— 5 }■ 
X 2 (1— cos /3)+2{cos (3 — cos w/3+cos (n— 1) (3—1 }] 
+2 -3 (l — cos/3) -3 [8 (1 — cos/3) 2 {cos/3— cosw/3+cos(w— 1)/3 — 1}-], . (18) 
which, when n(3=2c7r, 
= 2 -1 (1 — cos /3) -1 [(w 2 +2w+2) cos/3— n 2 — 4] 
— 2 -1 (1— cos /3) -1 [(2 w+6) cos /3— (2w+6)— 2] 
+ 2 -1 (l — cos/3) -1 [4cos/3— 4] (19) 
=2 -1 (l — cos/3) -1 [(w 2 +2w+2— 2w— 6 + 4) cos/3— w 2 — 4+2w+6+2 — 4], (20) 
=2 -1 (1 — cos /3) -1 [— n 2 (1 — cos/3)+2w] 
ri 2 n l 
" 2+2 
= cos /3 + 4 cos 2/3 + 9 cos 3/3+ +(w— l) 2 cos(w— 1)/3; j 
and as 0 2 cos 0/3 = 0, 
“-™- 1 n 2 n 1 
S m 2 cosm/3 =-- 5 -+ 7} 3 , 
m=o * sin 2 ? 
( 21 ) 
. (21a) 
except when (3 is 0 or a multiple of 2i r, in which case 
2 = ”~ m 2 cos m0 = 0 2 + 1 2 + 2 2 + 3 2 + +( w -l) 2 ~-|+| (216) 
Collecting together equations (36), (3d), (9a), (96), (11a), (116), (17a), (176), (21a), 
and (216), we have, according as /3 is not or is 0 or a multiple of 2t, 
2 sinm/3=0, 
771 = 0 
771 = 77 — 1 
2 cos m/3=0, 
or =0, 
1 a n B 
m sin m/3 =— -cot g, 
or =0, 
£ mcosm0= — = 
