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XIII. On the Calculus of Symbols . — Second Memoir. £yW. H. L. Russell, Esq., A.B. 
Communicated by Aethue Cayley, Esq., F.B.S. 
Made up of two Memoirs : one received January 7, — Head January 30, 1862 ; the other 
received June 18, — Eead June 19, 1862. 
This Memoir is a continuation of one on the Calculus of Symbols which I had the 
honour to lay before the Society in December 1860, and which has since been published 
in the Philosophical Transactions. I commence this paper with some extensions of 
the method given in the former memoir for resolving functions of non-commutative 
symbols into binomial factors. I then explain a method, analogous to the process for 
extracting the square root in ordinary algebra, for resolving such functions into equal 
factors. I next investigate a process for finding the highest common internal divisor 
of two functions of non-commutative symbols, or, in other words, of finding if two 
linear differential equations admit of a common solution. After this, I give a rule for 
multiplying linear factors of non-commutative symbols, analogous to the ordinary 
algebraical rule for linear algebraical factors. I then resume the consideration of the 
binomial theorem explained in the former memoir. Two new forms of this binomial 
theorem are here given; and the method by which these forms are proved identical 
wall, I hope, be considered an interesting portion of symbolical algebra, and as exhibit- 
ing in a remarkable manner its peculiar nature. 
In the next place, I proceed to calculate the general values of the coefiicients which 
occur in the form of the binomial theorem given in the first memoir ; I then obtain an 
expression for the symbolical coefficient of the general term of the multinomial theorem 
as pre\iously explained ; and also a theorem for the multiplication of symbolical factors 
emanating from each other after a given law ; lastly, I investigate a binomial theorem 
reciprocal to the binomial theorem already considered. 
In the former memoir I explained a process by which the symbolical function 
could be resolved in all possible cases into factors of the form 
I shall now give a method by which the same symbolical function may be resolved into 
factors of the form 
By pursuing methods similar to those employed in the preceding paper, we find the 
following equations as the condition that + i^aay be an internal factor of 
WDCCCLXII. 2 M 
