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II. On the Solution of Linear Differential Equations. By Charles James Hargreave, 
Esq., B.L., F.R.S., Professor of Jurisprudence in University College, London. 
Received June 10, — Read June 17, 1847. 
I. General Theorems of the Calculus of Operations. 
If the operation of differentiation with regard to the independent variable x be 
denoted by the symbol D, and if ^(D) represent any function of D composed of 
integral powers positive or negative, or both positive and negative, it may easily be 
shown, that 
(p(D){'<^x.u}—-<px.(p(D)u-{-'>p'x.(p'(D)u-\-^ffx.<p"{D)u-\-^4'"'x.(p'''(D)u-\-.. . (1.) 
and that 
(px.■<l.{D)u=■<l^(J)){(px.u}-^l.\D)ffx.u}+lff{D){fx.u}-^ff'(D){f'x.u}-\- . . ( 2 .) 
and these general theorems are expressions of the laws under which the operations 
of differentiation, direct and inverse, combine with those operations which are de- 
noted by factors, functions of the independent variable. 
It will be perceived that the right-hand side of each of these equations is a linear 
differential expression ; and whenever an expression assumes or can be made to 
assume either of these forms, its solution is determined ; for the equations 
(p(D){'^x.u} — F and (px.'4'(D)u=P 
are respectively equivalent to 
The formulae (1.) and (2.) indicate true propositions whenever they are interpret- 
able ; that is, whenever ip(D) and -4/(0) are capable of being expressed in integer 
powers of D. In conformity with recognized principles of reasoning, when the 
subjects of the process are regarded merely as symbols, we may assume that these 
propositions are true generally ; and we shall therefore not hesitate to pronounce 
any interpretable result derived from the free use of these theorems true, although 
the intermediate steps of the process are not capable of a rational interpretation. 
Bearing these remarks in mind, it will be seen, by an inspection of the above 
equations, that if in (1.) D be written for x, and —x (or t) be written for D, we obtain 
D' denoting the operation This equation is identical in form with (2.), and is 
