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V. On the Calculus of Symbols, icith Applications to the Theory of Differential Equations. 
By W. H. L. Eussell, Esq., A.B. Communicated by Arthue Cayley, Esq., P.B.S. 
Eeceived December 20, 1860, — Eead January 24, 1861, 
The calculus of generating functions, discovered by Laplace, was, as is well known, 
highly instrumental in calling the attention of mathematicians to the analogy which 
exists between • ditferentials and powers. This analogy was perceived at length to 
involve an essential identity, and several analysts devoted themselves to the improve- 
ment of the new methods of calculation which were thus called into existence. For a 
long time the modes of combination assumed to exist between different classes of symbols 
were those of ordinaiy algebra ; and this sufficed for investigations respecting functions 
of differential coefficients and constants, and consequently for the integration of linear 
differential equations, with constant coefficients. The laAvs of combination of ordinary 
algebraical symbols may be divided into the commutative and distributive laws ; and the 
number of sjnnbols in the higher branches of mathematics, which are commutative with 
respect to one another, is very small. It became then necessary to invent an algebra of 
non-commutative symbols. This important step was effected by Professor Boole, for 
certain classes of symbols, in his well-known and beautiful memoir published in the 
Transactions of this Society for the year 1844, and the object of the paper which I have 
now the honour to lay before the Society is to perfect and develope the methods there 
employed. 
For this purpose I have constructed systems of multiplication and division for func- 
tions of non-commutative s)'mbols, subject to the same laws of combination as those 
assumed in Professor Boole’s memoir, and I thus arrive at equations of great utility in 
the integration of linear differential equations with variable coefficients. 
I then proceed to develope certain general theorems, which will, I hope, be found 
interesting. I have applied the methods of multiplication, as just explained, to deduce 
theorems for non-commutative symbols analogous to the binomial and multinomial 
theorems of ordinary algebra. 
Lastly, I have shown how to employ the equations deduced in the earlier part of this 
paper in the integration of linear differential equations. I have, for this purpose, 
made use of methods closely resembling the method of divisors which has so long been 
used in resolving ordinary algebraical equations. The whole x)aper will, I hope, be 
found to be a step upwards in the important subject of which it treats. I shall just 
observe, that the symbolical combinations used in this paper may also be applied to the 
calculus of finite differences, as may be seen in Professor Boole’s memoir. 
MDCCCLXI. L 
