MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
25 
wherein 
_V-l+ s/{l'-lf-4U" 
“ 21 ’ 
0 
21 
^ m'—3m+ V {m! — m] 2 — 4mm" ^ m! —3m — V (m 1 — inf — 4mm" 
2 m 
2m 
n n —5n+ in' — n)^—4nn" 0 u n ' — 5n— V (n — «) 2 — 4m" 
a, — j p = ' • 
2 n 2 n 
It may happen that some of the factors, D-|-a, D+(3, &c., are wanting-. This would 
modify the investigation. We shall, however, here suppose that they are all retained, 
and shall seek the conditions of integrability under this supposition. 
The equation is reducible to the first order, and therefore integrable, if any of the 
following eight conditions is satisfied : 
« or l3=a r or (3' = a" or f3". 
Thus if we have a = «' = «", we find 
/(D -}- ,(3) u -f- m (D + fi') + n (D -f /3") z 26 u — (D + a) ~ 1 0 = ce ~ aS , 
which is an equation of the first order. 
Suppose that we have 
|3'=«"=/3"+l/| 
a! ={3 =a — l.J 
The equation then becomes 
/(D+a)(D+« — \)u-\-m(D-\-a— l)(D+f3')s^w+w(D+/3')(D-f-/3'— \)z 2t u=0\ 
(54.) 
or 
I D + a stl_r Z (D + «)(D + « — 1 
which has already been integrated. Ex. to Prop. 1, C. § 1. There are several other 
cases in which this method of reduction will apply. 
If the conditions (54.) are not both satisfied, let it be supposed that the first is 
satisfied, we have 
7Y) 71 
(D -T°0 (D-|-/3) w -]-y(D — os ) (D -j- (E -j-^f) (D-b^ — 1 )s 2 ^m=0. 
Put (D+/3'y=£, D=tt+^, 
m . n 
then 
(D + a)(D+/3)w+-^(D + «')^+y£ 2 M=0. 
Expanding the coefficients as directed in the rule, we have 
7TL 7)2 
(D+«)(D+(3) = (7r+«)(T+/3)+5(29r+a+/3— 1)^+5 2 ^ 2 . y(D +a')=y(T+«' + ?f ) ; 
the substitution of these values will give 
(5r+a)( 5 r+/3)M + |^29+y^w+ 9 (a+/3— 1) (? 2 +7?+7)f 2 ^— 0; 
or, as we may for simplicity write, 
(- r-j-a ) (x+/3)M-f- (A7r-j-B)gu-j-Cg 2 u=0, 
