MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
227 
basis, would be of peculiar character. Its actual advantages I conceive to be the 
following: — 
1. The necessary reductions, transformations and developments are effected, for 
the most part, by theorems, the expression of which is independent of the forms of 
/oW>/iW» &c - 
2. We are thus able to establish a perfectly general method for the solution of 
linear differential equations total and partial in series, and for the calculus of gene- 
rating functions. 
3. The form of the analysis affords facilities, which are believed to be peculiar to 
itself, for the finite integration of linear equations, and for the classification of in- 
tegrate forms. • 
The received theory of the solution of linear differential equations in series is given 
by Euler*. It consists in assuming u=z'Ha rn x m , and determining by substitution the 
relation connecting the successive values of a m , or as it is called, the scale of the 
equation. This method fails when, in seeking the first index of a development, we 
arrive at equal or imaginary values. I am not aware that any mathematician has 
shown how this failure is to be remedied. Now the method developed in this 
paper has no such cases of exception. 
The theory of series and of generating functions has been successively discussed by 
Euler and Laplace. A full account of their researches is given in Lacroix’s larger 
treatise on <the Calculus, tom. iii ., in the chapters Thdorie des Suites and Thdorie 
des Fonctions Generatrices. I class these investigations together, because, although 
their objects are distinct, their mathematical theories are virtually the same. Euler 
proposes to determine the generating function of a series, Hu m t m , when the coefficients 
are formed according to such a law as the following : 
u r 
(am + b)(a ] m + b x ) . . 
' (cm + e)(c 1 m + ej . . . 
u„ 
He shows that by successive differentiations and integrations, the factors am-\-b, 
cm-\-e ... may be eliminated, and the problem finally reduced to the solution of a 
differential equation. Laplace -f~, considering the unknown quantity u x in an equa- 
tion of differences as the general coefficient of the expansion of a function u, 
proposes to determine u , and then by expansion to obtain u x . It is not necessary 
for us to consider here whether the theory of generating functions is of any im- 
portance to the solution of equations of differences. The discovery of the gene- 
rating function of a series is in itself a problem both interesting and important. 
Those who have paid attention to the subject will, I think, admit that the theories by 
which Euler and Laplace have endeavoured to accomplish this object, labour under 
two defects, one arising from the tedious character of the process by which the dif- 
ferential equation is formed, the other from the difficulty of its integration. This 
does by no means derogate from the genius or the claims of those wonderful men ; 
* Calc. Integ. vol. ii. 
t Thdorie des Probabilites. 
