746 
PROFESSOR BOOLE ON THE. DIFFERENTIAL EQUATIONS WHICH 
C being a constant, the value of which does not change with i. Hence we may write 
and in like manner 
>J2=iD+^-iT- , =cr(^=iD+=') > 
” | _ n ' n J \ n 1 n y 5 
>J2_?_i\ = c'r(£--'). 
” \n n J \nn) 
The legitimacy of the introduction of T depends upon the condition 
n— 1 . , m _ i m _ 
®+-> 0 , > 0 , 
n ' n ’ n n 5 
so that the value i— 0 is inadmissible, as we have already assumed. Moreover m must 
lie between the limits — (n— 1) and 1 . 
Since e 6 =x, the equation for v is equivalent to 
d n v n 
d^+ v ~ 0 ’ 
whence 
v=c l e“ lX -\-c 2 e v . . -\-c n e* nX , 
in which a 2 , . ,a n are the nth roots of —1. This value of v can be expanded in 
ascending powers of x in the form 
v = v 0 -f- v x x -\-v 2 x 2 -f- &c. 
=^ 0+^1 e 8 +v 2 e 20 +&c. 
Hence u — u 0 representing that part of u which contains positive and integral powers 
of x , we shall have 
fn — 1,. m\ -r, /D m\ , . 
“-“.=CCT(— D+-)r(---)( B -* c) . 
Now 
v—v 0 = C^e 011 * — 1 ) + C 2 (<e^ — 1 ) . . + C re ( e* n * — 1 ) 
=2Q(^-1), 
the summation extending from i— 1 to i—n. Hence, merging CC' in the arbitrary con- 
stants C„ . . C n , we have 
“=“.+SC i r(^ D +f)r(?-f)(^-i), ...... (3) 
in which x=e e . This expression we now propose to interpret by definite integrals. 
Now e H * — 1 = I afi aih dJi. 
J 0 
Substituting and merging a t in the arbitrary constant Q, we have 
.=«,+sqr(5=! D+|)r 
=m 0 +SQ I dse~ s s n 
, r*“ D m 
J dte-'i »"»' 
t 
e H dh 
