276 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
F. § 1. Theory of Equations of Finite Differences. 
d 
r— a 
( fl. 1 \ ^ 
If we assume *r= — — - — 4 £ = drs dx , then it is shown (A. § 1.) that the following 
relations are satisfied : 
f{v)f l u=ff(v+m)u i (93.) 
(94.) 
and that 
c., (95.) 
the interpretation of ~ here being f{yr)=f(x) — /(w- 1). 
d d 
If r~— 1, we have tt=x—x g=xi~dx, and the above relations are still satis- 
fied. These values of t and § we shall first employ. 
Prop. 3 . Every equation infinite differences of the form 
X 0 «G + X y U x — i . . . -)— \. n U x — n~— X, 
X.jXjX, &c. being rational and integral functions of .r, may be reduced to the form 
J off u * J cf\ (fgUx +y &C. = U#, (96.) 
For multiply both sides of the equation by x(x—\). .(x — rc+1), and in the second 
term of the first member for xu x -\ write gu x in the third term for x(x— 1 )u x -2 write 
d 2 u x , &c., we shall have a result of the form 
%(x)u x + <pfx)zu x + <p.fx)£u x . .+<Pn(x)tUx=Ux 3 (97-) 
wherein <p 0 (x)—x(x— 1). .(x— n-\- 1)X 0 , <pfv)=(x— 1). .(x— w+l)X l5 . . . 
\Jjjjj^=.x(x— 1) . .(x— w+l)X, being all rational and integral functions of x. Now 
d_ _d_ 
— \—z~dx\ g=xs dx } whence x=vr-\-g. It only remains then to develope <p 0 (x), 
Pi(x), &c. in ascending powers of g. Writing then x for nfg in the first member of 
(95.), we have 
f( x )=ffK)+^f(v) § ±^-^f(v)f+ &c., (98.) 
the interpretation of ^ remaining as above. 
u= qihi t ^( K t ^~‘ 2 ^■■(^^ t —n + Z S jjy^.dx l ..dx n <p(a l —x„ a a _—x^..a n —x n ), 
the limits being given by the inequality 
on 2 -y> *2 of 2 
I ”2 I —/o 
V- + V“ + V 2< 
This solution requires that n should be odd. If n — 3, the result is equivalent to Poisson’s, but is in a form 
perhaps more convenient for physical deductions. 
