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VI. On the Calculus of Symbols. 
By William Spottiswoode, M.A., F.R.S. 
Made up of two Memoirs, one received November 4, Eead November 21, 1861, the other received 
January 21, read January 30, 1862. 
In a paper published in the Philosophical Transactions for 1861, p. 79, Mr. W. H. L. 
Ktjssell has constructed systems of multiplication and division for functions of non- 
commutative symbols, subject to the same laws of combination as those in Professor 
Boole’s memoir “ On a General Method in Analysis,” Philosophical Transactions, 1844. 
In this calculus there are four systems of multiplication and division, viz. internal and 
external, both (1) when the functions are arranged in powers of (2) when in powers 
of X, or, as they may be termed, the ^-arrangement, and the ^r-arrangement. In the 
paper in question the author has confined himself, so far as division is concerned, to 
the case most useful in practice, in which the divisor is linear. And of this he has dis- 
cussed in full only the g>-arrangement. 
§ 1. Internal division of the x-arrangement by a linear factor. 
Adopting the same notation as Mr. Russell, I propose here to investigate, in the first 
place, the condition that '4'i(^)7r-}-'4/o(?} internal factor of 
and to determine the quotient. This is partially discussed in pp. 73-75. 
Let 
then performing the actual divisions, for brevity writing for •v^'(f), and p for (p(§), 
Hence the condition that ^aay be an internal factor of 
will be 
«.(?)-?>,(?) t-^=o 
( 1 -) 
