500 



tions of the symbol D, which, for the sake of simplicity, is written in 

 place of — . 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, trans- 

 formations and reductions are immediately effected by theorems the 

 expression of which is independent of the forms of the functions 

 /q (D),/*! (D), &c. Accordingly it may be shown that various for- 

 mulae which have been given for the solution of linear differential 

 equations, with those in which Laplace's theory of generating func- 

 tions 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 differ- 

 ential 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 inte- 

 gral : 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. 



The finite solution of linear differential equations may be attempted 

 by resolution of the proposed equation into a system of equations of 

 an inferior order. This method applied to the linear equation under 

 its usual forms leads to the well-known solution of equations with 

 constant coefficients : and when applied to the same equation under 

 the exponential form, it gives a result embracing the solution not 

 only of equations with constant coefficients, but also of a large 

 class of equations with variable coefficients. 



The author treats, — 1st, of the solution of linear differential equa- 

 tions, total and partial, in series ; 2ndly, of their finite integration ; 

 Srdly, of the theory of series, or inverse method of development ; 

 4thly, of linear equations of differences, total and partial, of certain 

 miscellaneous applications, chiefly in the field of definite integrals, 

 single and multiple. 



January 25, 1844. 



SIR J. W. LUBBOCK, Bart., V.P., in the Chair. 



" A Description of an extensive Series of the Water Battery ; with 

 an account of some Experiments made in order to test the relation 

 of electrical and chemical action which takes place before and after 

 completion of the Voltaic Circuit." By John P. Gassiot, Esq., F.R.S. 



In a former paper, which was printed in the Philosophical Trans- 

 actions for 1839, the author described a series of experiments made 



