814 



2. " General Methods in Analysis, for the resolution of Linear 

 Equations in Finite Differences and Linear Differential Equations." 

 By Charles James Hargreave, Esq.. LL.B., F.R.S. &c. 



The investigations presented in this paper consist of two parts ; 

 the first offers a solution, in a qualified sense, of the general linear 

 equation in finite differences ; and the second gives an analysis of 

 the general linear differential equation with rational factors, so far 

 as concerns its solution in series. 



The author observes that there does not at present exist any ge- 

 neral method of solviug linear equations in finite differences of an 

 order higher than the first: and that with reference to such equa- 

 tions of the first order, we obtain insufi^cient forms which are intel- 

 ligible only when the independent variable is an integer. It is in 

 this qualified sense that the solutions proposed in this paper are to 

 be taken : so that the first part of these investigations may be con- 

 sidered as an extension of this form of solution from the general 

 equation of the first order to the general equation of the ?^th order. 



In tlie second part, the author points out a method by which the 

 results of the process above indicated may be made to give solutions 

 of those forms of linear differential equations whose factors do not 

 contain irrational or transcendental functions of the independent 

 variable, or contain them only in an expanded form. 



This object is effected by means of the theorem, relative to the 

 interchange of the symbols of operation and of quantity, propounded 

 by the author in a former memoir published in the Philosophical 

 Transactions (Part I. for 1848, p. 3i). It is one of the properties 

 of this singular analytical process that it instantaneously converts a 

 linear equation in finite difierences into a linear differential equation ; 

 so that whenever the former is soluble, the latter is soluble also, 

 provided the result be interpretable ; a condition satisfied when the 

 functions employed are rational algebraical functions. 



Notwithstanding the qualified character of the solutions previously 

 obtained forlinearequations in finite differences, the solutionsobtained 

 from them by this process are free from all restriction. The solutions 

 in series can be written down at once from the equation itself, inas- 

 much as each series has its own independent scale or law of relation ; 

 and no difficulties arise from the appearanceof equalorimaginary roots 

 in the equation determining the incipient terms of the series. These 

 circumstances do indeed cause a certain variation of form; but they do 

 not compel us to resort to any special process in each individual case. 



The perfect separation and independence of the scales, or laws of 

 relation of the series enables the author to discuss the characters of 

 the series with reference to their convergency or divergency, and to 

 classify these equations into sets having peculiar and distinguishing 

 properties in regard to this subject. 



The first set includes those equations whose solutions can always 

 be found in convergent series oi ascending powers of the independ- 

 ent variable ; and if in such case the equation be solved in series of 

 descending powers (which can be done by this process), those series 

 are certainly always divergent. 



