232 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
and by induction, 
(g- 1 7r) n u=g- n T(<?r — 1 )..( tt — nf \)u, 
riff — ] )..(“?r — w-j- 1 )m = '|^ 
f - r — 1 71 
J 1 — £ dx I 
whence 
But 
wherefore 
u. 
u. 
t(tt— l)..(sr — n-\-\)u=^ n y 
A 
g n =x(x-\-r)..(x-\-(n — 1 )r)z v ' dx , 
f s dx — 1 ] 
nff — \)..(jr—n-\-^)u=x[x-\-r)..{oc-\-{n — l)r)| — - — j \ 
=x(x+r)..(x+(n— l)r)(^) w. 
Scholium. In the values of tt and § employed in the two last propositions, if we 
A 
expand the exponential A dx , we find 
* =J? (s+r 3ns?+iis'^+ &c -)’ 
? =x (l+r* +hi?+ kc A 
Let r— 0, then r=x-^, g=x. Put x=z e , then f=e*. For simplicity, let us 
represent ^ by D, then by (II.), (III.), (IV.) the symbols D and g* satisfy the follow- 
ing relations : 
f(D)z m6 u— z m f\ D+m)M (V.) 
f(D)i m6 =f(7n) z m6 (VI.) 
D(D— 1)..(D — n-\-\)u—x n {^j u (VII.) 
These are known relations. With a view to the maintenance of an unbroken 
analogy, it has, however, been thought better to deduce them from the properties of 
the more general system in % and g, than to assume them as already proved. 
B. § 1. Theory of Linear Differential Equations. 
Prop. 1. Every linear differential equation which can, with or without expansion 
of its coefficients, be placed in the form 
(«+te+rf..)^+(®'+^+c'A.)^+& e .=X, 
may be reduced to the symbolical form 
/„(D)«+/ 1 (D)A+/ 2 (.D) S “«...=U ) (VIII.) 
wherein f Q , /), f 2 ... are functional symbols, and U is a function of g*. 
