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MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
for the value of every discovery is in some measure relative, and is to be measured 
by the state of contemporary science as well as by its abstract merit. 
The advantages which the method of this paper is believed to possess as respects 
the theory of series, are the following : — 
1. The law of the series being known, or the equation of differences given, the 
differential equation is known by inspection. The rule is absolutely general, whatever 
may be the forms of the coefficients. 
2. The form under which the differential equation is presented offers great facili- 
ties for its integration. Those facilities are chiefly owing to the circumstance, that 
the form of the equation permits us, as before remarked, to effect the requisite trans- 
formations by general theorems. That this form has a peculiar fitness for the pro- 
cesses of integration, is further shown by the circumstance, that the method of reso- 
lution which in the common theory leads to the solution of differential equations 
with constant coefficients, conducts us here to the solution of a large class of equa- 
tions with variable coefficients. 
The arrangement of the subjects treated in this paper will lead us to consider, — 
1st, linear differential equations; 2nd, the theory of series; 3rd, the theory of 
generating functions ; 4th, the theory of equations of finite differences. 
A. Preliminary Theorems. 
Prop. 1. Let w and § be distributive symbols which combine in subjection to the 
law 
§A*)u=>f{*)eu, (1.) 
being a functional symbol operating on in such manner that \f(<r)z=f(jp(‘r)\ it 
is required to expand in ascending powers of g. 
We have 
§f(*)u=>f(*)§u, 
§ 2 f{v )u=X 2 f(v)g‘ 2 u, 
Let ;r -]-£>= * 7 , then f(n-\-g)u=f(ri)u. Now, as operates solely on u, it is commutative 
with respect to the constants in wherefore 
nf{n)u=f(n)w. 
Or dropping the subject u , and writing v-\-g for y, 
(^+M^+£)=/( T +£)( T +e)- 
Let then, still supposing u to be understood, 
fa+d/fc + f) 0)£ m + glf m (v)g m , 
= ^rf m {n)g m + 1gf m {n)g m , 
= 2^„(T)g" + 2^ m (a-)£*+ 1 by (2). 
