MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
247 
From the former of these equations we have 
(D + ^iq — qfD-\-a)s?u v —0, 
(D -\-b)u i — qf(Y5 1 ) w — 0, 
(*S + b ) u i-<h x ( x 7L +«+l)«i=0, 
{x-q^) -jf +(fe- (fl!+ l)^) Ml = 0, 
q _ q> 
x b [^.—qpc) a ~ b " rl ' > U - x b (l — q T v) a ~ b+1> 
q(l — q q x) a ~ b+1 +c 2 (l — q l x) a ~ b+1 
U x b (\—q- l x) n ~ b+l (\~q 12 x) a ~ b+l 
The same process would solve the same equation with a second member X. 
The next class of equations to be considered comprises those which are integrable 
by a transformation operating on the dependent variable. 
As the theory of the general equation 
u + <p x (D) + p 2 (D) • • • + <P n (D ) z n *u = U 
is deducible from that of the equation 
u-\-<p(D)z r hi = U, 
we shall first consider the simple case. 
Proposition 2. — The equation u-\-<p(J})i r6 u~\] will he converted into the form 
-j- 0 ( D -j- ri)z r6 v — V, by the relations 
u = i n6 v, \J = £ n ^V (XIII.) 
For assume u— i n6 v, and substituting in the original equation, we have 
z n6 v -j- <p ( D)g( w + r )A> = U, 
.*. s n6 v -j- z n f{I) n) i r6 v = U, by (V.), 
v -{- <p (D -j- n) t r6 v = s ~ L 
Let 6-^U=V, then ll=s nf V, and the above becomes 
v (p(J} J r n)z r6 v=Y , • 
as was to be shown. 
Proposition 3 . — The equation u-\-<p(D)s ri u=-U will he converted into the form 
v-\-^(J})z r6 v—Y , hy the relations 
u- 
■p v 
' ^(D) 5 
it — p ZilZ y 
wherein denotes the infinite symbolical product p^p y-)q,(0 — ^r) " ' ' ’ 
For assume u=f(J))v, and substituting in the original equation, we have 
/(D) y +9(D) g ^D) i ;=U, 
2 K 
MDCCCXLIV. 
