1864.] Mr. W. H. L. Russell on the Calculus of Symbols. 423 



" On the Calculus of Symbols. — Fourth Memoir. With Ap- 

 plications to the Theory of Non-Linear Differential Equa- 

 tions.^' By W. H. L. Russell, Esq., A.B. Received July 31, 

 1863"^. 



In the preceding memoirs on the Calculus of Symbols, systems have 

 been constructed for the multiplication and division of non-commutative 

 symbols subject to certain laws of combination ; and these systems suffice 

 for linear differential equations. But when we enter upon the consideration 

 of non-linear equations, we see at once that these methods do not apply. 

 It becomes necessary to invent some fresh mode of calculation, and a new 

 notation, in order to bring non-linear functions into a condition which 

 admits of treatment by symbolical algebra. This is the object of the fol- 

 lowing memoir. Professor Boole has given, in his ' Treatise on Difi'erential 

 Equations,' a method due to M. Sarrus, by which we ascertain whether a 

 given non-hnear function is a complete differential. This method, as will 

 be seen by anyone who will refer to Professor Boole's treatise, is equivalent 

 to finding the conditi5ns that a non-linear function may be externally 

 divisible by the symbol of differentiation. In the following paper I have 

 given a notation by which I obtain the actual expressions for these con- 

 ditions, and for the symbolical remainders arising in the course of the 

 division, and have extended ray investigations to ascertaining the results 

 of the symbolical division of non-linear functions by linear functions of the 

 symbol of differentiation. 



Let F (x, y, 2^3 • • • • y») ^^^Y non-linear function, in which 



2/3) • • • • 1/n denote respectively the first, second, third, .... nth 

 differential of y with respect to (x) . 



Let denote fdy^, i. e. the integral of a function involving oc, y, y^, y^»,.» 

 with reference to y,. alone. 



Let V,. in like manner denote — when the differentiation is supposed 



dy, 



effected with reference to y,. alone, so that V,. U^F=F. 



The next definition is the most important, as it is that on which all our 

 subsequent calculations will depend. We may suppose F differentiated 

 (m) times with reference to y„, y„_i, or y„_2> &c., and y„, y„_i, or y^-n 

 &c., as the case may be, afterward equated to zero. We shall denote 

 this entire process by Z^;"\ Z^,% Z,[% &c. 



The following definition is also of importance : we shall denote the ex- 

 pression 



d d . d , d , , d 



by the symbol Y,. 



* Read Feb. 11, 1864 ; see Abstract, vol. xiii. p. 126, 



VOL. XIII. 2 I 



