MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
241 
Should T 0 involve ri, x", &c., the operations indicated may be of a kind which it is 
impossible in the present state of analysis to perform. In some such instances they 
may be evaded by a linear transformation, and in all cases the difficulty will be placed 
in the true point of view, — no slight advantage of the method. 
The theory of equations involving a second member is, mutatis mutandis, the same 
as explained in the preceding section. 
Ex. 1. Let the equation be such, that, on assuming x=z\ we have 
(D-a)(I)-b)u+<p(y,-^’ D)s"w=0. 
Here, by the fundamental theorem, 
u = 2u m i m6 = '%u m x m . 
(m— a)(m — b)u m +<p(y } ^ m)u m _ x — 0. 
The equation (m—a)(m—b)=0 gives m=a, m=b; and as arbitrary functions are to 
be written in the place of constants, we shall have 
u=¥ a {y)x a +F a j r \{y)x a + l +¥(a+‘r ) {y)x a +‘ i . . . 
- \-Yb{y)x iJ r¥ b j r \{y)x b + l -\-¥ b j r 2 {y)x b + 2 . . . 
where F a (y), F b(y) are arbitrary, and in general for the rest 
m ) 
Fm(j/) — a) (ni — b) F m ~ 1 ^y) • 
(p u (ffiw 
As a particular example, let =f(y)-^. 
Multiply by x 2 , and putting x=z 6 , 
d 2 
D(D-l)w-/(y)^s 2 'M = 0, 
“=F 0 (y)- 
1.2 ' 1. 2.3.4 
1. 2.3.4 
d 2 
r 1 r d- l 
M^ x {y) \f{y)^\?Ay) 
+F 1 (y>+ 5 
d?_ 
df 
1.2.3 ** ‘ 1.2.3.4.5 
d 2 
F 0 ( 3 /)F 1 (?/) being arbitrary. As the operation, implied by the symbol f{y)~yyp can 
always be performed, the above solution is universally interpretable. 
if/C$/)= a2 > we s et 
« 2 d*F 0 (y) 
a 4 d 4 F 0 (y) 
or 
M — F o (y) + 1>2 dy 2 X 1. 2.3.4 dy 
+ t l\y) x ~r 1.2.3 dy 1 ^ 1.2.3.4.5 dy 4 " 
Put F 0 (y)=<p(y)+^(y), F \{y)—u{<p\y)-^(y)), and substituting, we get 
u — <p(y + ax) + My — ax ) , 
which verifies the solution. 
2 i 2 
