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XIII. General Methods in Analysis for the resolution of Linear Equations in Finite 
Differences and Linear Differential Equations. By Charles James Hargreave, 
Esq., L.L.B., F.R.S., Professor of Jurisprudence in University College, London. 
Received March 15, — Read March 29, 1849. 
Preliminary Remarks. 
1. The investigations presented in this paper consist of two parts ; the first offers 
a solution, in a certain qualified sense, of the general linear equation in finite differ- 
ences ; and the second will be found to give an almost complete analysis of the 
resolution in series of the general linear differential equation with rational factors. 
The second part is deduced directly from the results of the first, although the 
subjects of which they respectively treat appear to be wholly independent of each 
other. 
With the exception of a few cases capable of solution by partial and artificial me- 
thods, there does not at present exist any mode of solving linear equations in finite 
differences of an order higher than the first ; and v/ith reference to such equations of 
the first order, we are obliged to be content with those insufficient forms of functions 
which are intelligible only when the independent variable is an integer, and which 
may be obtained directly from the equation itself by merely giving to the indepen- 
dent variable its successive integer values. It is in this insufficient and qualified 
sense that the solutions here given are to be taken ; and the first part of the follow- 
ing investigations may be considered as an extension of this form of solution from the 
general equation of the first order to the general equation of the wth order. 
Linear Equations in Finite Differences. 
2. A complete analytical theory of the general equation of the wth order, 
would involve its resolution into a series of equations of the first order of the form 
i^x—P'fx-l = G'', 
1 / =CU^‘ 
