MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
239 
the solution whereof by the fundamental theorem, is 
A =2<i s mi , B = 2A z md , 
m 2 a m +2mnb m +pm a m _ 1 +pnb m _ l + (q(m 2 —n 2 )-\-r)a m _ 2 +2qmnb m ^=0, 
m2 b m — 2mna m +P m b m -i—pna m _ 1 -{-(q(m 2 -ri i )+r)b m _ 2 -2qmna m _ 2 =(). 
Whence making i 6 —x, and determining the values of a m , b rn from the two last equa- 
tions, we have 
u— cos (n log^)(a 0 + fl 1 ^+a 2 .T ,2 4- &c.) 
4- sin (n\og x)(b 0 -\-b l x J rb 2 x 2 ~\- See.) 
a 0 and b 0 being arbitrary, and the remaining coefficients formed on the laws 
_ p{nP + 2n' 2 )a m _ l — pmnb m _! + {q(m 2 + SrP) + r)ma m _ 2 + 2n(qrP—r)b m _. 2 
a,n m(m 2 + 4ra 2 ) 
, pim 2 + 2n l )b m _ l +pmna m - l 4- (g(?« 2 + 3n 2 ) + r)mb m _ 2 — 2n{qn 2 —r)a m _ 2 
u m m(ni 2 + Arv 2 ) 
If q—0, y— 0, r— 1, we have for the primitive equation 
and for its complete integral, 
u— cos ( n \ogx){aQ-\-a 2 x 2 -\-a i x ‘ i! . . .) 
+ sin ( n log# )(b 0 -\-b 2 x 2 -\-b i x 4: . . .), 
where in general 
__ ma m ^ 2 — 2nb m _ 2 , ^ mb m _ 2 + 2 na m _ 2 
a m m{m 2 + 4/A) ’ m m{m 2 - \-4ri 1 ) 
The above developments terminate in convergency for every value of x. The 
more general developments (16.) from which they are derived, become, in certain 
cases, divergent, as is seen by making m infinite in the equations determining a m , b m . 
The descending developments which are then to be employed may be thus obtained. 
We have 
(D 2 -f n 2 ) u - j- (qD 2 +r) z 26 u =0. 
Multiply by z~ 26 and invert the order of the terms, then 
( 9 (D + 2) 2 -fr)M+p(D-l-2)£-^+((D+2) 2 +w 2 ) g - 2 ^=0. 
Put 6=— and the above becomes 
(< 7 (D — 2) 2 -j-r)w — p{ D — 2)z 6 m-\- ((D — 2) 2 +ra 2 )g 2 ^M=0. 
The equation (^(D — 2) 2 +r)w=0 determines the form of the general solution, 
which will differ according as — shall be positive, negative, or 0. The process is in 
all respects the same as in the preceding examples, except that in the result we shall 
have s*i= ~zr- 
2 i 
MDCCCXLIV. 
