238 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
whence m 2 a m -\-2mb m -^-a m _ y —0 for all values of m , and m 2 b m +b m _ y =0 for all values of 
m except m— 0, which gives m 2 b m -\-b m _ 1 =\, or b_ x = 1 ; also from the other equation, 
a_!=0. From these, the values of a m , b m , corresponding to negative values only of 
m, may be determined ; whence writing x for z\ and solving the above equations rela- 
tively to a m _ y and b m _ yi we have 
„=«_,+ ^+^ + &c. 
+ log x (b_ y + b -^ + &c.), 
where 
«-i= 0, &_!=] ; 
and in general 
a m -i=~ ( m 2 a ni -\-2mb m ), b m _ y ——m 2 b m . 
This is a particular integral ; to complete the solution we must add the general 
value of u given by the equation ^ -\-xu= 0, which, as in the preceding ex- 
amples, is found to be 
a^a^x-^-afl 2 . . . 
+ log x (&0 + V+ V 2 • • •)> 
wherein a 0 , b 0 are arbitrary constants, and the succeeding coefficients given by the law 
_ rna m _ 1 — 2b m _ 1 __ _ K-\ 
m 3 ’ m ffi 2 
(]*">! (Ill 
Ex. 4. Given (xt+qofl) 2 -\-(x-\-px 2 -\~bqx‘ i ) ^ -\-{n 2 ~\-px-\-{Aq-\-r)x 2 )u=0. 
On assuming x—z 6 , we get 
(D 2 -\-n 2 )u-\-pT)z 6 u-\- (qD 2 -\-r)z 26 u = 0, (15.) 
which, as a final example, we propose to integrate both by ascending and by de- 
scending developments. 
The equation D 2 m+w%= 0 gives u=A cosw0-f-B sin n0 ; substituting this value in 
(15.), and regarding A and B as parameters, we have 
(D 2 ~\-n 2 )u= cos w0(D 2 A+2wDB)-|- sin w$(D 2 B— 2wDA) 
pT)z 6 u— cos n6(pDz 6 A-\-pnz 6 R) + sin nQ(pD^B~ pm 6 A) 
qD 2 z 26 u= cos n6(^ — qri 2 z 2e A-\- qT) 2 z 26 A-\-2,nqT)z 26 )i) 
sin nd{ — qn 2 z 26 B>-{-qjy 2 z 26 Q — 2nq~Dz 2( A) 
rz 26 u = cos (n&)rz 26 A-\- sin {nO)rz 26 B. 
Collecting these results, and equating to 0 the aggregate coefficients of cos n& and 
sin n6, 
D 2 A-f2wDB-{-p(D£ ( ’A+ns^B) -f- (g(B 2 — n 2 ) -\-r)z 26 A-\-2nqDz 26 Ji=0, 
D 2 B — 2»D A -\-p(Dz 6 A — wsT») -j- (q (D 2 — n 2 ) -f- r)g 2 *B — 2nqY)z 26 A=0 ; 
