260 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
wherein A=2g-{-y> B = 5'(a+j3— l)+ya', C = q 2 -\-jq-\-~- 
This is integrable in several distinct cases. 
1st. Put C = 0, and taking either value of q, suppose that we have A = 0, B=0, then 
( ‘7T -J-Gt) (ft -\-(3 )u=0, 
which is reducible to a pair of equations of the first order, and is completely inte- 
grable. 
2nd. Determine q so as to satisfy the equation C=0, then 
(t -f- a) (n + (3) u -f- ( At - j- B)^ m = 0, 
B 
,r + A 
or M + A ( r +a )(»+|3r = 0 - 
g 
This is reducible to the first order whenever ^ differs by an integer from u or (3. 
g 
Thus, suppose that —a is an integer, then assuming 
B 
we have u= Pi|^»,=P 1 ^«',= (»+f)(»-+j- l)..(»+*+l)». 
The equation for determining v is of the first order, and the derivation of u from v is 
effected by processes which involve differentiation only. 
3rd. Let A=0, and determining q, let B=0, then 
+ <*){% + U 
Suppose a greater than (3, and assume 
then w=P 2 ,r ^“ p V — (r+u— l)(5r-f-«— 3)..(<r+(3+2)v. 
Here it is necessary that a and (3 should differ by an integer. The equation deter- 
mining v is reducible to a pair of equations of the first order, and u will be found, as 
in the last example, by differentiation ; thus, 
w=(D —qg)v — {D— gfD-f (3')z*}v, 
= {D— qs*(D+p'+l)}v, 
= q(p+l)xv. 
If in the general equation (53.) we assume and multiply by s 2 ^, we find 
?A(D'-a"-2)(D'-|3"-2)M+m(D'-a , -2)(D , -|3 , -2)/w + /(D , -a-2)(D , -^-2)£ 2 ^=0. 
