752 
PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
Now 
therefore, finally, 
g(2 (»- 1) + l)ir V- 1 _ g(2n-l)«V- 1 _ g-*V- 1 
^(2tt+I)nV-l _ gif'/— 1 • 
^=l+ 2 ^ v ’j* dv v n 'log^l — e n>J 'xV^—e n v 'J*. dv v n 'log 
It is seen, from the form of this expression, that it represents a real value. 
1 
If we substitute v for v n , a change which does not afifect the limits, there results 
u=y m = l + 0 ™ 1 | dv fl m_ 4og(l — ■— dv 'y m ~ 1 log^l — e 7i ^~ 1 
in which V = ^ - n • This expression we shall now reduce to an equivalent real form. 
(l-e^-'xV^ 
xV 
Reduction of the expression for y m . 
9. We shall somewhat simplify the general expression above found for y m by inte- 
grating by parts. The integrated portion will be found to vanish at both limits. 
Representing ~ by V', we have 
log xY\ dv=v m log (f— e 1 *' / '~ 1 xV') +#e"’ r '' /-I P — v Vdv 
Now, expanding the logarithm in the integrated portion, and putting for V its value 
— — , we see that that portion will consist of a series of terms of the form 
1+0 r 
J± v m+(n-l)r 
(l+«") r ’ 
r being for each such term a positive integer. 
All these terms vanish when ir=0, since, by the conditions to which m is subject, 
m-\-(n— l)r is positive. 
Again, they vanish when v is made infinite, since in this case 
p^ v m+{n-l)r 
( 1+0 
=Av n 
and, by the conditions relative to m, the index m—r is negative. 
We have, then, on applying: the above reduction to the terms of the general value 
oitf 
"V’dv 
■xNe~n ^- 1 
f - 
Jo l . 
n V'dv 
■xVe^-'J 
