1863.] 579 



Hydrochlorate of Rosaniline ............ C 20 H 20 N 3 Cl. 



C H N Cl 



Hydrochlorate of Triphenylic Rosaniline . . " 3 



(t-'e H 5) 3 - 



Details of these experiments I hope to lay before the Society at 

 an early meeting. 



VII. " On the Calculus of Symbols." Third Memoir. By W. 

 H. L. RUSSELL, Esq., A.B. Communicated by A. CAYLEY, 

 F.R.S. Received May 15, 1863. 



(Abstract.) 



In my second Memoir " On the Calculus of Symbols," I worked 

 out the general case of multiplication according to one of the two 

 systems of combination of non-commutative symbols previously given. 

 In the present paper I propose to investigate the general case of 

 multiplication according to the other system. I commence with the 

 Binomial Theorem, to which the second system gives rise. In my 

 previous researches I obtained the general term of the binomial 

 theorem when the symbols combine according to the first system 

 by equating symbolical coefficients ; here, on the other hand, I con- 

 sider the nature of the combinations which arise from the symbolical 

 multiplication, and obtain the general term by summation. I next 

 proceed to the multiplication of binomial factors. Here the general 

 term is obtained by considering the alteration of weight undergone 

 by certain symbols in the process of multiplication. The multi- 

 nomial theorem according to the second system is next considered 

 and its general term calculated. I conclude the memoir with some 

 applications of the calculus of symbols to successive differentiation. 

 This paper completes the investigation of symbolical multiplication 

 and division according to the two systems of combination, the 

 general case of division having been worked out by Mr. Spottiswoode 

 in a very beautiful memoir recently published in the Philosophical 

 Transactions. 



The Society then adjourned over the Whitsuntide Recess to 

 Thursday, June 11, the President having announced the Meeting 

 for the Election of Fellows to take place on Thursday, June 4, at 

 4 P.M. 



