226 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
d / \ du 
s («»)=« s . 
</ / , N du . dv 
These properties of the symbol — , taken in connection with the principle above 
enunciated, lead to the most important results that have been yet established by the 
calculus of operations. We have an early example of their application in the sym- 
bolical form of Taylor's theorem, viz. 
f{x+h)=6 h tef(x). 
A result to which we shall often refer is the following-. If we have a linear equa- 
tion with constant coefficients of the form 
■K n u J \ r \ l Ti n ~ l ' u ' J r A 2 flr M ~ 2 M . . . -|-A n w=X, 
wherein tt operates solely on «, and is therefore commutative with respect to A 15 A 2 , 
&c., then 
u= {T n -\- A^ -1 -}- A 2 7t w ~ 2 . . -j-A w } _1 X 
=N 1 (tt— a 2 ) _1 X+ &c., 
N l5 N 2 ..a l5 a 2 .. having the same values as in the resolution of the rational fraction 
f. n , A yr \ 7-r- into a similar series of terms*. 
£ + • • ~T 
It is obvious that the above method is of necessity limited in its application. It 
is only in linear equations with constant coefficients that the operative symbols com- 
bine in subjection to the law we have supposed. Accordingly it has been remarked, 
that the calculus of operations has tended rather to simplify the processes of ana- 
lysis than to extend its power. 
The object of this paper is to develope a method in analysis, which, while it ope- 
rates with symbols apart from their subjects, and may thus be considered as a branch 
of the calculus of operations, is nevertheless free from the restrictions to which we 
have alluded. The linear equation with variable coefficients, whether in differentials 
or in finite differences, will be treated under the form 
/o(*> +/i (**)£“ +/ 2 (^)£ 2 “ + &c. = U, 
U being a function of the independent variable x, and r and operative symbols, 
which combine in subjection to the law 
= ei‘i K -I- rn) u, 
and which, when the subject function u is unity, further satisfy the relation 
/(7r)^=/(m)^. 
It might be expected, a priori, that a theory of linear equations founded on such a 
* Cambridge Mathematical Journal, vol. ii. p. 114, vol. iii. p. 239. 
