266 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
A still more remarkable theorem is the following. 
Let u = 2 ffmpn 2 .. .m n )x l m cv 2 m i..xf n > the summation extending to all positive integer 
values (0 included) of m v m 2 ..m n , which satisfy the condition 
then if — x 2 , &c., 
u=f(D (64.) 
wherein 
Xo = 
(s 6 i — — e^)...{eh — s 6 n) 
s (n-\)6. 2 
2 ( £$2 — £^i)(a^2 — £^3)..(s^2 — e^n) 
and so on for the rest. What is particularly to be noticed is, that the quantities 
X l5 X 2 ..X ;i are independent of v. 
Lastly, if the condition under which the summation is to be effected is 
the rest as before, then 
r 1 .(»+l)^l 1 .(v-j-l)A, 
u — f(P\-> ^2-^jjXi \_ e g 1 + X 2 i_ s g 2 +&c. } . . . . (65.) 
As an example, suppose it required to obtain a finite expression for 2 (mnx m y n ) 
subject to the condition 
Here by the theorem, 
w=D i D ^{^:+l^r} 
m-\-n— v. 
Thus let v=3, we have 
as it evidently should be. 
dr a?*+l — ?/»+l 
' dxdy x — y 5 
__(v — 1 )(«*+!— j/ v + 1 ) + (v + l){xy—x' l y) 
~ ly 
2x A — 4x 3 y -( 4 xi/ 3 — 2y A 
u=ly (^F~ 
= 2x 2 y-\-2xy 2 , 
D. § 2. On the Theory of Generating Functions as connected with Equations of 
Differences. 
The complete solution of the equation of differences 
u m ^Qi( m ) u m-i + %{™)u m _ 2 ... + <pfm)u m _ n =f(rn), .... (66.) 
involves n arbitrary constants. This implies that n successive values of u m may be 
regarded as indeterminate, the remaining values being thence formed according to 
the law of which (66.) is the expression. The research of the generating function of 
u m implies the finding of a function u, developable in a series, tu m x m , of which the 
first index p, and the first n coefficients u p , u p+1 ..u p+n _ 1 are arbitrary, and the remain- 
ing coefficients are formed in subjection to the law (66.). 
