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XIV. On the Calculus of Functions. By W. H. L. Eussell, Esq.^ A.B. 
Communicated hy Aethur Cayley, Esq., F.B.S. 
Received October 31, — Read December 5, 1861. 
One of the first efforts towards the formation of the Calculus of Functions is due to 
Laplace, whose solution of the functional equation of the first order, by means of two 
equations in finite differences, is well known. Functional equations were afterwards 
treated systematically by Mr. Babbage ; his memoirs were published in the Transactions 
of this Society, and there is some account of them in Professor Boole’s Treatise on the 
Calculus of Finite Differences. A very important functional equation was solved by 
Poisson in his Memoirs on Flectricity ; which suggested to me the investigations I have 
now the honour to lay before the Society. 
I have commenced by discussing the linear functional equation of the first order with 
constant coefficients, when the subjects of the unknown functions are rational functions of 
the independent variable, and have shown how the solution of such equations may in a 
variety of cases be effected by series, and by definite integrals. I have then considered 
functional equations with constant coefficients of the higher orders, and have proved that 
they may be solved by methods similar to those used for equations of the first order. I 
have next proceeded with the solution of functional equations with variable coefficients. 
In connexion with functional equations, I have considered equations involving definite 
integrals, and containing an unknown function under the integral sign; the methods 
employed for their resolution depend chiefly upon the solution of functional equations, 
as effected in this paper. The Calculus of Functions has now for a long time engaged 
the attention of analysts ; and I hope that the following investigations will be found to 
have extended its power and resources. 
Let the functional equation be 
+2n 8 —a<p(x}=F{w), 
where <p is an unknown, and F a known function. 
Let 
n 2 
+- COS 2, 
2r r ’ 
and the equation becomes 
or if 
