MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
237 
whence we find 
m3 b m + ( l b m-n+6m 2 c m =0, 
rn 3 c m +qc m _ n =0, 
q 
Wl d m — n 377lbm—n~ 1 “ m—n 
mr 
b _ =-i 
mhm—n 6Cm—n 
mr 
Cm—n 
C m = 7 3 
m—n, 
7tl 3 
(14.) 
If we now substitute the preceding values of A, B, C in the equation w=A+B0-fC0 2 , 
and then change s* into .r, d into log x, we shall have 
u=(a 0 + a n x n +a 2u x In ...) 
+ log x(b^ -j- b n x + b % x ln . . .) 
+ (log x) 2 (c 0 +c n x n + c 2n x 2n . . .), 
a 0 , b 0 , c 0 being arbitrary, and the succeeding coefficients determined by (14.). 
The solution of the linear differential equation U=X is found by obtaining a 
particular integral, and adding to this the complete integral of the equation U=0. 
A particular integral of the equation U=X will be given by the fundamental 
theorem whenever X is developable in ascending powers of x. If X is of the form 
Xo+X.log <r+X 2 (log^) 2 ... +X n (loga?) n where X 0 , Xj.. are respectively developable 
in powers of x, we must assume u= A-f-BA. +P0", where A, B...P are variable para- 
meters, to be determined by the fundamental theorem, in the forms A = 2 aj m6 , 
B = 2b t m6 , See. 
On the same principle we must proceed if such forms as cos (wlog#), sin (wlog#), 
&c. are found in the second member. 
Ex. 3. Given .r 2 ^ +x ^+xu — log {x). 
Putting x= r J, we have 
Y) 2 u-\-z 6 u=6. 
Make w=A+B$, then on reducing 
D 2 A + 2DB + s^A -p (D 2 B + — 1 ) 6 = 0, 
whence, as in preceding examples, 
D 2 A + 2 DB-H<A = 0 , 
D 2 B+g'B = l. 
This system of equations differs from those before considered, in that the second 
members do not both vanish. The fundamental theorem gives 
A = 2a m z m6 , B = 2b m t mi , 
2{(m 2 fl m +2jw6 w +a m _ 1 )e w# } = 0, 
2{(rn 2 & m +& m _,) i mS } = \, 
