258 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
Substituting the expanded forms of f Q (D) j\ (D) &c. in (52.), we have a result 
which may be thus represented : 
?()(«■)« + ? , i(’ r )^+?>>(’*')f 2 +&c. = U, (XVI.) 
p 0 (5r), <Pi(k), &c. being rational and entire functions of it. Several distinct cases may 
here occur, rendering the above equation either integrable, or reducible to a simple 
form. 
1st. It may happen that by a particular determination of n the equation (XVI.) 
may be reduced to a single term. Suppose that it should give 
<Po (lt)u=U ; 
then if it— t— & c - are the factors of <p 0 (w), we shall, by resolution, have 
u = Njzq + N 2 w 2 + N 3 w 3 + &c. 
O'— gi)^i=U 
O'— ? 2 ) w 2= u - 
On replacing it by its value, D—n<p( D)g^ the above system will assume the form 
M+<p 1 (D)g^=U, 
which has already been considered. 
2nd. The coefficients <£> 0 (it), ^(ir), &c. may be constant. The equation then be- 
comes 
u -f - a x gu + ci.g 2 u -j- &c . = U, 
which has been already considered. 
3rd. The equation (XVI.) may, perhaps, be reduced to consist of a pair of terms. 
Suppose that it should give 
?oW M + , ?i( 5r )? M =U ; 
this may be reduced to the general form 
u-\-<p(it)gu=\J~. 
As it and § combine according to the laws of D and &*, the above equation may be 
treated by Prop. 2 and 3, C. § 1, and its integrable cases determined accordingly. 
In illustration of the above theory, we shall investigate the principal integrable 
cases of the equation 
d^u du 
( lx 1 -j- mx 3 -f- nx i ) -j- (Fx-\- m'x 2 -f- n'x 3 ) ^ + ( /" -f- m"x -(- n"x 2 ) u— 0 . 
The symbolical form of the above equation is 
{/D(D — 1)+/ / D-|-/"}m- 1- {m(D— 1)(D — 2)+m , (D— 1 ) + m " } & 6 u 
-f- {w(D — 2)(D — 3)+« , (D — 2)-\-n"}s 2 ^u = 0 ; 
or, as it may be written, 
l(D + «)(D+Q)u+tn(V + d)(D+@yu+n(D + a")(I)+p")v u u=0, . . (53.) 
