MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
257 
D + f. — 2 
, b b 1 9 » n * 
v-\— - pt ; 7i ie v=Li~* 
' a (D+71— 1) ' a D + n — \ ’ 
and putting i 6 —x and reducing, 
dv An— \)a + (f— b)x+gx z C 
dx' ax + bx 1 V x 2 (a + bx ) 5 
an equation of the first degree, whence the value of v being determined, that of u is 
given by (50.). 
The methods above developed are also applicable, mutatis mutandis, to partial and 
to simultaneous equations. 
C. § 2. Some Illustrations of a more General Method of Resolution. 
We have, in Propositions 2 and 3 of the last section, fully considered the theory of 
the differential equation M+<p(D)s , %=U, and have exhibited in Prop. 1 a method of 
resolution by which a more extensive class of equations may be reduced to the pre- 
ceding form. In what follows I purpose to exemplify a more general method of reso- 
lution, founded on the expansion of /(^r+f) in (I.). This method is deserving of par- 
ticular attention for two reasons; first, because, in connexion with Propositions 2 and 3 
already referred to, it enables us to integrate almost every class of linear differential 
equations that admits of integration in finite terms ; and, secondly, because a strictly 
analogous method is applicable to equations of finite differences. 
I shall suppose the differential equation to be placed under the form 
fo (D)m +/ L (D)p(D) i e u -\-f ,(D)<p(D ) <p(D — 1 ).«u . . . =U, 
where f Q ( D), /^(D), &c. are any rational and entire functions of D, and <p(D) any 
function whatever of that symbol. 
Let D — w<p(D)g*=sr and <p(D)s g = §, then by reasoning precisely similar to that of 
A, Prop. 3, it is seen that % and £> combine according to the law, 
%ff)u=ff-\)%u. 
Now <p(D)e*=o, and soon. Wherefore the proposed equation 
will assume the form 
/o(D)M+/ 1 (D)gM+/ 2 (D)f 2 M+ &c. = U (52.) 
But D =7r-\-n§, wherefore, expanding the coefficients of the above equation by (I.), 
we have 
/o(D) =/„(*) +«^/oWf + r 2 + &c - 
And similarly for the rest, the interpretation of ^ being 
=/(**) “ 1 ) • 
2 l 2 
