Mr. W. H. L. Russell on the Calculus of Symbols. 439 



I. 



I now proceed to apply the calculus of symbols to the solution of func- 

 tional equations. 

 Let 



dx 



Then the following formulae are known : 



(e'^WS)y(,^■)=/{x-•(x.^•+2)}. 

 &c. = ... 



(e'^'^'S)./(.t.)=/{x-'(x(.v) + '•)}• 

 These formulee may be thus expressed in the notation of the calculus of 



symbols : if p=e^'^^^ 7r=x, d a functional symbol acting on /(7r)in such 

 a manner as to convert/(7r) into /x~'(x^+ > ^^^^ 



a general law of symbolical combination due to Professor Boole. 



We will now consider two cases of internal and external division in which 

 the symbols combine according to this law. The results, as will appear 

 afterward, will be found useful in the solution of functional equations. 



And, first, for internal division. We shall determine the condition that 



+ may divide |o'*0„ (tt) + jO^-i(/;;j_ i(7r) + The process will 



be, mutatis mutandis, the same as in my former memoir. The symbolical 

 quotient is 



and the required condition is found by equating the symbolical final re- 

 mainder to zero, and we have 



6 affecting every part of the term which succeeds it. 



I shall now give the corresponding condition for an external factor. The 

 symbolical quotient is 



The required condition is found by equating the final remainder to zero ; 

 we have 



V^|7r -Jy^TT U/^TT — xLl^TV Xp^TT xL^TT \f/^7r 



0-1 in each term affecting everything which comes after it. 

 I conclude with some examples of functional equations. 

 Let the functional equation be 



VOL. XIII. 2 K 



