250 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
The symbolical form of this equation is 
T 
D(D-l) .. + ( 32 ‘) 
/ d\~ n 
wherein V is the symbolical form of ( X, i. e. the result obtained by writing d 
for x in the ??th integral of Hdx n , no constants being added in the integration. From 
inspection of (32.), it is evident that the class of equations sought, must, on assuming 
x — z s , be reducible to the form 
ll 1 /]TV j \7n j \ / TV j \ — T_J • 
(D + « 1 )(D -+ a 2 ) . . (D + a n ) 
in which we shall suppose the quantities a v a 2 , . . a n to be ranged in the order of their 
magnitudes. Put u=s- a ^u } then by Prop. 2, 
u. 
D(D + a 2 - Cl ) . . ( D + a n - ai ) ^ u \~^ (33.) 
The first factor of the denominator of <p(D) in (32.) now corresponds with the first 
factor of the denominator of -^(D) in (33.). In any of the remaining factors we may 
by Prop. 2 convert D into B+/r, i being any integer, — hence that they may all cor- 
respond with the factors of ^(D). we must have the quantities 
a„ — a i + l a 3 — ffj + 2 a 4 — a^ + 3 a, l — a 1 + n— 1 
n 5 n ’ n n 
all negative integers, which are therefore the conditions sought. 
From (32.) and (33.), by (XIV.), 
?q = P; 
$(D) 
wherein f(D) =± D(D + ^(P) = + D(D _ 1) .("(D-r. + i) i 
but u=e~ a i i u l , wherefore 
(D — 1) .. (D-ra+1) 
(34.) 
qD-j-flg — a \ )»»(D + o re — <q) 5 
whence the value of u will be deduced from that of v by differentiation ; for since 
a 2~ «i< — V 
P Vd?^^) = ( D- ^(D-w- 1 ) •• ( d +« 2 — «i + w )> 
and so on for the remaining factors to which P„ is to be applied. 
The two following examples will sufficiently illustrate the preceding case. 
Gw 
Ex. 2. Given ^j z -\-q 2 u — ^- = 0, which is an equation occurring in the theory of 
the earth’s figure. 
The symbolical form of this equation is 
,,2 
u 
J_ 1 ,26,. — A 
' (D + 2) (D — 3) 6 U ~~ y} ‘ 
