433 Mr. W. H. L. Russell on the Calculus of Symbols, 



we shall find the above equations satisfied ; and consequently the last in- 

 vestigation gives the law of the formation of the remainders. Each re- 

 mainder will of course be subject to the three conditions already exhibited. 



These results point out the foundations on which symbolical division, 

 as applied to non-linear functions, must rest. We have confined our at- 

 tention to external division, as more particularly applicable to these func- 

 tions. When a non-linear equation is proposed for reduction, we must 

 ascertain whether it admits of an external factor by employing the method 

 of division as already explained. 



On the Calculus of Symbols. — Fifth Memoir. With Application 

 to Linear Partial Difi'erential Equations^ and the Calculus of 

 Functions.^' By W. H. L. Russell, Esq., A.B. Communicated 

 by Professor Stokes, Sec. R.S. Received April 7, 1864"^. 



In applying the calculus of symbols to partial differential equations, we 

 find an extensive class with coefficients involving the independent variables 

 which may in fact, like differential equations with constant coefficients, be 

 solved by the rules which apply to ordinary algebraical equations; for 

 there are certain functions of the symbols of partial differentiation which 

 combine] with certain functions of the independent variables according to 

 the laws of combination of common algebraical quantities. In the first 

 part of this memoir I have investigated the nature of these symbols, and 

 applied them to the solution of partial differential equations. In the second 

 part I have applied the calculus of symbols to the solution of func- 

 tional equations. For this purpose I have given some cases of symbolical 

 division on a modified type, so that the symbols may embrace a greater 

 range. I have then shown how certain functional equations may be ex- 

 pressed in a symbolical form, and have solved them by methods analogous 

 to those already explained. 



we shall have 



OP, omitting the subject, 



also d d d d 



X- y- — \-x +y —X +y' -\-x~ y ; 



dy ^ dx ^ dy ^ dx 



therefore the symbols ~y ^ x^-Yy"^ combine according to the 



laws of ordinary algebraical symbols, and consequently partial differential 

 * Read April 28, 1864. See Abstract, vol. xiii. p. 227. 



