MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
277 
Developing by this theorem the coefficients of (97.), the result will assume the form 
/o(« , )«*+/i(«‘)^+/ 2 (* , )f 2 *<*+ &c. =Up, (XXL) 
as was proposed to be shown. We shall call this the symbolical form of the equation. 
To revert from the symbolical to the common form, it is only necessary to observe 
that since 7r=x— §, and since x and § combine according to the law gxu x —(x — 1 )gu <• 
we have fff) -f(x) - ^ X f{x)g -f^/O* 1 ) ^ - &c., . . . .(XXII.) 
wherein Ax= — 1, and consequently ^/(.r) = t /(.r) — /'(.r— 1). Developing by this; 
theorem the coefficients, and writing for g m u x its value x(x — 1). the 
required reversion will be effected. 
F. § 2. On the Solution of Equations of Finite Differences in Series. 
In the equation fff) u x-\~f\ (fgu* • • • -\-fr{^)fu x =-0, 
assume u x —'2u m g m then precisely as in the case of differential equations, 
/o (m) u m -f/i (m) u m _ i . . . +f r (m)u m -r=0 (XXlil.) 
The initial values of m are determined by the equation f 0 (m) = 0. For every such 
value u m is an arbitrary constant, for all other values it is successively determined 
by (XXIII.) 
Ex. 1. Given (x— a)u x — {2x— a— \)u x -\ + (1 — q 2 )(x— l)w*-2=0. 
The operation at length stands thus : 
Multiply by x, we have 
x(x — a)u x — (2x— a— 1),zm#_i + (1 — q 2 )x(x— l)u x -2=0. . . . (99.) 
Or x(x—a)u x —(2x—a—\)gu x -\-(\—q 2 )fu x =0 (100.) 
Now developing the coefficients by (98.), we find 
x(x— a) = t(t — a) + (2 w — a — 1 , 
2 x—a— 1 —2 T—a— 1 -|-2f, 
whence substituting in (100.) 
— a)u x — q 2 ffu x — 0, 
wherefore u—'%a rn g m , with the relation 
m(m — a) a m — q 2 a m _ 2 = 0, 
01* Cirri — 
Q Cl m — 2 
’m(m — a) 
The equation m(m — a ) =0 gives 0 and a for the lowest values of m, wherefore, finally. 
r(] -I g 4 g(a?- 1 )(g-2)(g-3) ■ » \ 
M .r- L V 1+ 2.2 -a '+ 2.4(2 — a) (4 — a) +&C -'V’ 
, n ( , lW , ^ ■ gMg-l).-(g-a-l) - q 4 x(x-l)..(x-a-3) g \ 
])..(x — «+!)+ 2(2 + a) 2.4(2 + a) (4 + 
