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 differences; 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 methods, there does not at present exist any mode of solving linear equations in finite differences of an order higher than the first; and with 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 independent 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 following 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 with order.
Teaching GLM Concepts: Explicating the Connections
