256 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
By reasoning precisely similar to that of Prop. 3, it may be shown that the equation 
. . . 4-<p w (D)g^M=U may be converted into the form v . . . . 
+ %//»(D)s n *y=V, by the assumptions, 
_p 
it — p jhijjjy _p 
V 1b ^(D) 
(XV.) 
That these assumptions may be realized, it is necessary that ^(D) . . ^(D) 
should be so chosen as to satisfy the conditions, 
4> a (P) _ 4>,(P)4>i(P-l) 
^ 2 (B) 'I'i(D)4'i(D — 1)’ 
4i n (P) -_ ^i(D)^i(D — 1) . -<h( D — » + 1) 
4'«(D)~^ ] (D)^i(P — 1) • • 'I'dD—tt+l) 
These conditions being satisfied, the first of the equations (XV.), viz. z/ = P 1 ^ 1 | ^| u, 
\rill enable us to deduce a from v. 
It is seldom that an application of the above theorem is necessary, and a single 
example on the present occasion will suffice. 
Ex. 7- Given (a+bx) e ^+(f+gx)^+ngu= 0. 
The symbolical form of this equation is 
a U 
Assume as the transformed equation, 
b + b 
D + — — 2 
, b b t,g D + « — 2 9J 
u-\-— s hi J r — tvtv — — s 26 u — 0. 
a B(D — 1) 
1 ? + — . — 7 ~ TTl~ -s 2 ^=V. 
a D + n — 1 1 a \J + n — 1 
Here we have 
»i(P)_ P 
p ^(P) p 
*2*8(0) — *2 
- 1 = (D+«-I)(D+n-2)..(D + l), 
<Pt ”g(D-i) , '~ 2) = ( D +' i - l)(D+«-2) . . (D + l 
), 
and these forms are identical. Hence 
w=(D+»— l)(D+» — 2). . (D+l)u=(^) x n ~ l v, .... (50.) 
0 = (D+w— l)(D + n-2)..(D+l)V (51.) 
As a factor of <p 2 (D) has disappeared in the transformed equation, it is necessary, by 
the second canon, to retain an arbitrary constant in the value of V. Now the com- 
plete integral of (51.) is \ — cz~ 6 -\-c \- 16 . . +c"£-(”~ 1 )‘ , 3 whereof we shall retain the first 
term in the second member. Hence 
