736 
PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
if we give to l n the particular value 1, we have 
rm+(n—l)r 
r 
n [r] 7 
and the expansion then represents the mth power of that particular root which, when ‘ 
#=0, reduces to 1. The law of the series upon which the formation of the differential 
equation depends is, as we shall perceive, independent of these determinations. 
Changing r into r — n, we have 
rm+(n— l)r 2=T +1 
_“L — n~ n J xi - 
whence the law of the series is seen to be 
(6) 
and therefore, by what is shown in my memoir “ On a General Method in Analysis,” the 
differential equation defining u will be 
[d >+ [^M— i]r , (5_5_ 1 )^ =0 , ... . (i) 
in which e e =x and D=~ • 
3. As u may here represent the mth power of any of the roots of the given equation, 
it is evident that the general integral of the above differential equation will be 
w=C$r+C 2 ^.- + (7) 
exception arising, however, in the case in which for a particular value of m the n parti- 
cular integrals y y%, . . y™ cease to be independent. In such cases the above value of u 
constitutes an integral, but not the general integral of the differential equation. 
For instance, if m= — 1, and if we reduce the given algebraic equation to the form 
(y~ l Y+xy~ l - i=o, 
it is evident that, except when n= 2, we shall have 
1= 0- 
Here then 
n=C 1 yT'+ C^-h.+C ^" 1 
may be reduced to the form 
\u=(C- CJyf 1 + (C 2 - Q n )y~' . . +(C„_-C n )y~l„ 
virtually involving but n— 1 arbitrary constants. 
Such cases of failure may, however, be treated by giving to the integral a form which 
for the particular value of m shall become indeterminate, and then seeking the limiting 
