254 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
Ex. 5. Given (1 -^ 2 )^ 4 - (^±i_(4w-^+l)^^—2w(2w-jo)w=0. 
This equation has been discussed by Poisson *. I am, however, unacquainted with 
his results. 
Passing to the symbolical form, we have 
u 
(D + 2» — 2)(D + 2rc — 2 — p) 0 . 
V) " 
which is integrable in three distinct cases. 
D(B +p) 
e 2 *u=0, (42.) 
1st. Whenp is an odd integer, by assuming as the transformed equation 
(D+2rc— 1— jo)(D+2n — 2— p) oe 
17 (D+/>)(D+^~1) -z-v-0, 
then operating by (XII.). 
2nd. When n is an integer, by assuming 
D + 2n 2 —p 0 
v ~ — u+p — *'»=*> 
which is of the first degree. 
3rd, When 2 n—p is an even integer, by assuming 
v— z- 6 v—\. 
An equation similar to the above, and susceptible of an interesting physical appli- 
cation, will be treated at length in another part of this paper. 
We are now prepared to assign the general conditions of integrability of the equa- 
tion U-\-<P\P)i r6 U—\} . 
In the first place, if <p(D) involves factors of the form , in which V -~ - is an 
integer, they may be made to disappear as in the two last examples. 
Such factors being then rejected, let the remaining factors, if any, in the numerator 
of <p(D) be (D-H>qKB-bm 2 )..(D+w r ), and the remaining factors, if any, in the de- 
nominator of <p(D) be (D+w 1 )(D-f-w 2 )...(D+w».). The conditions required are, that 
the quantities. 
m 2 ~ m i + 1 ni 3 —m 1 + 2 m r —m } +r — l~ 
ry y 
n 9 .~ ra i + l w 3 — Wj + 2 n r — n^ + r— 1 
(43.) 
shall be all integers. 
For in such cases the proposed equation can, by (XIV.), be transformed into the 
following : 
T-p /(D)/(D — 1)..(/D — r-p \)s, r6 v=-Y, (44.) 
m, , 
wherein /(D) is equal to (D+wq), or to (D-P/q)- 1 , or to p according as the fac- 
tors of <p(D), under consideration, are of the form (D-p«q)(D-pwi 2 ). .(D+m,.), 
or 
_ (D + w t )(D + m 2 ). ,(D + m r ) 
(D + j?j)(D + ra 2 ). .(D + n r ) ’ ^ (D + ?q)(D + w 2 ).,.(D + n r ) 
* Journal de l’Ecole Polytechnique, call. xvii. p. 614. 
