236 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
Now this is the type of the system of relations derived from the term f 0 (D)u. But 
the equation y* 0 (D)M=O is satisfied by the assumption w=AP+BQ... in which A, B.. 
are constants, that is, by the assumption w=a 0 P+6 0 Q.., whence the lowest value of 
m in a m b m .. is 0, and the system (12.) is universally satisfied. 
ct^u du 
Ex. 1. Let the primitive equation be x 2 -^—(^{a-\-b—\)x-\-qx 2 ")-^- x -\-{ab-\-cqx)u~Q,. 
Putting x—z^, we have 
D(I) — 1 )u — (a-\-b~ 1 ) Du — q r JDu -\-{ab~{- cqz 6 ) u=0. 
Now qz ( Du=q(D — l)s 6 u by (V.), hence we have 
{D(D— 1 ) — (a-\-b— 1 )D + aZ/}'«— (y(D— 1 ) — cq)zhi=Q, 
or (D — a)(B — b)u — q(D — c— l)^u=0, 
which is the symbolical form of the equation, whence u = 'Lu m z m6 , with the relation 
{m — a) {m - b) u m - q (m — c — 1 ) u m _ , = 0, 
. (m—c—l)u, n _ 1 
whence u m — q\- w-f — rr- 
m i ( m — a){m — b ) 
The equation (m— a)(m— b )=0 gives m=a or b, which are consequently the lowest 
indices of the development. If, therefore, we represent the arbitrary constants u a , m 
by A and B, we have 
u 
.( , a — c „ (a—c)(a—c+ 1 ) „ \ 
=K^+ U(a-b+l) ^ + +9 1.2(a-b+l)(a-b+2) x++kc -) 
Ex. 2. Given ,r’’ ^77 -\- 3 x 2 ^ -\-:c jy -j r y.v’‘u = 0, to find u. 
Putting x—z 6 , we have by (VII.), 
{D(D— 1 )(D— 2 )+ 3 D(D — 1 )+D}m+ 9 £^w.= 0 , 
D 3 m -f qz n6 u — 0 
(13.) 
Now as D represents the equation B%=0 gives u— A-fB^+C^ 2 . Substituting this 
value in (13.), there results 
D 3 A + qz n6 A + (D 3 B + qs»*B ) 0 + 3 D 2 B + (D :? C + qz n *C ) d 2 + 6D 2 C0 + 6DC = 0. 
Whence by the rule 
DA + qz n 6 A + 3D 2 B + 6D C = 0, 
D 3 B + qz n 6 B + 6D 2 C = 0, 
D 3 C + ^C=0; 
wherefore by the fundamental theorem 
A—^ a m z m6 , B =2^, C= 2 c m z"A 
