750 
PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
the above value, and giving to sin its exponential form, we have 
-r-(2; + l)r T /_, -Qn-tQ-Cgj+l)^ , ^ 
% r sin 
Now in general 
2 v/-i 
mir v '-l„ ~2Q'+ !>"■ /_, m-7r V-l -2?Vir 
c ra Z r e » — e * 5 Ve « 
2 v' — 1 
g pkrrr ^—1 1 j ^2fcnV— 1 | ^nkit-J- 
e (n + \)kirj-l gfor //— 1 
gkir ij— 1 — J 
knit 
k(n+\)ir ,_j g 2 
=0 2 x- 
Putting therefore 
we have 
and putting 
. kmr 
*J2L+2W- 1 Sm ~ 2 ~ 
= e 2 x— IT' 
sm- 
n 
, ^ V-i — ~°' + y ” +l) ff ^- 1 sinQ' + l)^ 
sinU±i& 
*=«=£, 
Hence 
Now 
-V-x aKttiW-x sin jV 
sin>’ 
n 
2 r sin 
. / m— r \ _ r I 
m =2^-1 
C7 + l)(«+l)_ /_, • / • , ,\ 
— 1 sin (j + 1 )tt 
sin 
n 
-m-j(n+l) v 
sin (j+ l)ff n 
for all values of j taken from the series 1, 2, . . n except the value n— 1, for which the 
expression becomes indeterminate in form, and has for its true value 
ir cos ms n cos ms 
mr cos ti 
— +W, 
as n is odd or even. 
