748 
PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
but from the law of the series as expressed in (6), art. 2 
E+— ]'"*(-;) 
u n — 
Equating these values, 
u 0 ==m 
M* 
m\ 
n) 
, „T _1 
n\^+ n - 2 \ 
=2C, 
l m . vm 
— sin — 
n n 
by the reductions of art. 6. 
Hence, finally, , 
— sin — *^° \ 
n n ' 
which agrees with the previous result. 
(II) 
Determination of the Constants. 
8. I propose here to determine the constants of the general integral (II), so as to 
obtain an expression for the mth power of that particular (real) root of the equation 
y n —xy n ~ ’ — 1 = 0 
which becomes unity when x=0. 
We have 
m , 
«=2C«— +2C A dv log(l— a^V), ..... (1) 
m . mn J 0 ov ' ' ' 
-2cu 
where V = L_L , and a* represents in succession the different nth roots of —1. 
1 +v L 
The coefficient of x r in the expansion of this value of u in ascending powers of x will 
be found to be 
r ^ m+ (n — l)r ^ r 
T-r(r) 
and its coefficient in the expansion of y m by Lagrange’s theorem is, for the particular 
root in question, 
* ’ 9 
n[f] r 
2 c,«r=- 
m 
^m+(n—\)r ^ 
r-1 
nrt 
m+ (n— l)r^ 
! r l 
( r 
— m\ 
. - 
K n ! 
equating which we have 
