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In a paper published in the Philosophical Transactions for 1861, p. 79, Mr. W. H. L. Russell 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 ofρ , (2) when in powers of π, or, as they may be termed, theρ -arrangement, and the π-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 dis­cussed in full only theρ -arrangement.

VI. On the calculus of symbols