The purport of this paper is to exhibit a new form of analysis, and to found upon it a new theory of Linear Differential Equations, and of Generating Functions. The peculiarity in the form of the analysis consists in the linear differential equation, instead of being represented, as it has hitherto been, under the type X 0 d n u / d x n + X 1 d n -1 u / d x n -1 . . . . + X n u = X, X 0 , X 1 , &c. being functions of the independent variable x , being exhibited in the form f 0 (D) u + f 1 (D) ϵ θ u . . . . + f ϵ (D) ϵ r θu = U; in which ϵ θ = x , and f 0 (D), f 1 , (D), &c. imply functional combinations of the symbol D, which, for the sake of simplicity, is written in place of d / d θ . This the author calls the exponential form of the equation; and he, in like manner, designates the analogous forms of partial and of simultaneous equations. What he conceives to be the great and peculiar advantage of the exponential form, both as respects the solution of linear differential equations, and the theory of generating functions, is that the necessary developments, transformations and reductions are immediately effected by theorems the expression of which is independent of the forms of the functions f 0 (D), f 1 , (D), &c. Accordingly it may be shown that various formulæ which have been given for the solution of linear differential equations, with those in which Laplace’s theory of generating functions is comprised, interpreted into the language of the author, are but special cases of theorems dependent on the exponential form above stated, and which are susceptible of universal application. The common method of effecting the integration of linear differential equations in series fails when the equation determining the lowest index of the development has equal or imaginary roots. In a particular class of such equations of the second order, Euler has shown that log. x is involved in the expression of the complete integral: but this appears to be merely a successful assumption; and the rule of integration demonstrated in the present paper admits of no such cases of exception whatever.