372 MESSES. C. AND F. CHAMBEES ON THE MATHEMATICAL EXPEESSION 
and smi being a small angle, we may write for its sine smi, and for its cosine (1— |s 2 mV), 
when we obtain 
a m /= \_(p s cos smz+q a sin smz)-\-si(q s m cos smz—p s m sin smz) 
s 2 i 2 -i 
-g- (p s m 2 cos smz-\-q s m 2 sin smz) J . 
Multiplying both sides by cos tmz, t being any positive integer, 
a m , cos tmz—a m cos tmz-\- m cos smz cos tmz—p s m sin smz cos tmz ) 
“i 
— -£-(jp s m 2 cos smz cos tmz-\-q a m 2 sin smz cos tmz ) J ; 
and taking the sum on both sides from m = 0 to m=(n— 1), 
(17) 
. (18) 
m—n — 1 m=n— 1 
% a m , cos tmz— 2 a m cos tmz 
m=0 m= 0 
m—n — i r ^ ^ 
+ 2 j -x { q s {m cos(s -f £)m;S + m cos (s — £)to 2) —p s (m sin(s + t)mz + m sin(s — t)mz ) } 
m= 0 \_ Z j 
s 2 i 2 . i I 
— (p s (m 2 cos(s -j- t)mz -j- m 2 cos(s — t)mz) -f- q s (m 2 sin(s -(- t)mz -{- m 2 sin (s — t)mz) } J. J 
Now observing, from the collected equations at the end of the first set of demon- 
strations in the Appendix, that when nv=2cT, and according as v is not or is 0 or a 
multiple of 
m cos mv= — x. 
S m sin ww= — s cotg, 
m = 0 
or 0, 
«j2 
“-”- 1 nr n 1 nr n* , 
2 m 2 cos mv= — 0 + „ , or x — 77- {-*, 
2 1 2 . 0 3 2 1 6 
m-0 Sin 2 
> (20) 
% m 2 sin mv=— cot; 
or 0. 
J 
And as in equation (19) nz=2,cir, applying equations (20), equation (19) becomes 
s + ^ n s — t \ ) ‘'j 
w. 2 n 1_ 
~o"i 2 <?_ 
( n n' 
/ n 2 n 
)-?.( 
1 
“2+2 . 
sm 2 ~ z 
L s + t 
n 2 
Cot —o~Z- 
-9- cot 
(19) 
