744 
PKOEESSOB BOOLE ON THE DIFPEEENTIAL EQUATIONS WHICH 
above theorem, 
Accordingly 
But since 
we have 
Therefore, finally, 
r(.-i^ri-’)= 
mr\n- 
”- 2+ »] 
• /i m \ 
sin( 1 ) i 
\ n ) 
[”- 2+ 7t] 
71—1 
7 r 
sin | 
(;•: 
) 
p ■^■n 7t 
0 m . nrn 
— sin- — 
n n 
Nje 9 , N„e e Aje® • • + A n e nB 
1 -«/" ' 1 — ci n e 9 1+^® 5 
A re =(-1) (—■■ +^j ai ci 2 ...a n 
= (-!)” (B.-.+BJxC-ir 
= B 2 • • + B„ . 
p B 1 + B,.. +B re 
0 m . m% 
— sin — 
n n 
Substituting in (7), and replacing e° by x, we have 
l= &±lilL^L +B r dt fi-log(l-^T) +B,f“^ ^-■log(l-« J ^T), . (9) 
oin ^0 0 
wherein, it must be remembered, that a 1 , a 2 , . . cc n are the several nth roots of — 1, and 
ihr 
T= 
i + t 
And this is the general integral of (I), B x , B 2 , . . B„ being the arbitrary constants of 
the solution. 
Second Method.— 1 . For the finite solution of differential equations of the form 
/.( D>+/,(D)«*V=0, 
it is usually convenient to reduce them to the form 
