738 PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
which is satisfied by u=y~ l and by u—y^\ but, as is evident from its unhomogeneous 
form, not by «t=C 1 yf 1 +C a yj' 1 . In this case, in fact, the condition 0 not 
being fulfilled, the primary differential equation (I) suffers no change in the form of its 
general solution. 
Mr. Harley’s results are in effect transformations of (10) and (11). Since u=y~\ 
it is seen that u will satisfy the algebraic equation 
u n -\-xu— 1 = 0 . 
Transform this by assuming 
1— re 1-re 1 __1 
x= —n(A—n) n x! 71 , u=(l—n)~ n x? n u\ 
and we have 
v! n — mi -\-(n — 1 )#'=(), 
which is Mr. Harley’s algebraic equation in form. Hence, if ci—e 9 ’ and D f =^, we 
shall have 
e 9 = — n(l — n)~e~\ u=(l—n)~”e~ 6 'u', D=~— D'. 
And (10) will become 
Multiply by f( n and we have 
Now 
1 —n 
e (n-w u '-\_-B'+n-2] n -\-n) n -\l-ny- n u'=0. 
and 
Hence 
or 
[_n+rc— 2]»- i =(-i)”- i py]’ 1 - 1 . 
IW V -( 5 V 1 )*“[i=I D'-^]”V*->V=0, 
which is Mr. Harley’s equation (1), art. 1. When n= 2, we obtain from (11), by the 
same transformations, Mr. Harley’s second equation (2), art. 1. 
Not only for the particular value ra=— 1, but apparently for all integer values of m, 
the general differential equation (I) admits of one integration. It may be said that 
while the differential equation determining the form of the mth power of a root of the 
algebraic equation is in general of the nth order, this equation may, when m is an integer, 
be reduced to an equation of the n— 1th order; not, however, like the higher equation, 
