5 io SCIENCE PROGRESS 



which may be regarded as an equation for $ when / is given. The theory 

 of permutable functions, 1 a recent development of the theory of integral equations, 

 is based on the last equation. The calculus of functions has been developed 

 largely by Babbage, 2 J. F. W. Herschel, Abel, De Morgan, Boole, Leslie Ellis, 

 and other writers. Babbage uses the notation f[f(s)] =f 2 (s),f"(s) = f\f n ~ l (s)] 

 and raises the question as to the meaning of f"(s) when n is not an integer. 

 A similar question has been raised as to the meaning of D n y when n is not an 

 integer and when D denotes the operator d\dx or some other operator. A theory 

 of fractional differentiation and integration has indeed been developed by 

 Liouville, 3 Boole, 4 Riemann, 5 Heaviside, 6 and other writers, and is of considerable 

 interest in the present subject because it is closely connected with the theory 

 of integral equations. Liouville, in fact, uses definite integrals of the types which 

 occur in the integral equations of Abel and Fourier to define his fractional 

 operations, and it is easy to see how integral equations can be used to solve 

 equations involving fractional derivatives and vice versa. 



In like manner integral equations can be used to solve functional equations, 

 as the following example due to Leslie Ellis 7 will indicate : 



Solve the equation 



/ <t>m{x)<pn(a - x)dx = cf) m + re (a), where <j> n (x) is an even 

 function oix. 



Assume ty„(x) = / Cosav. (j>„(a)da, then 



tym(x)tyn(x) =ff Cos ax Cos $x. QJflft H (p)dad& 



=yy "Cos (a + ®X. <t> m (atf>„{p)dadtt 



=J f Cos yx. <l>m{a)<t>n(y - a)dady 



= / Cos yx. (p m + „(a)da — yjf m + „{x). 



Hence ty.„{x) = [x(x)] m , where \( x ) ]S independent of m. Inverting the equation 

 \x(x)] m =f*CosaX. <$> m {a)da 



1 V. Volterra, The Theory of Permutable Functions, Vanuxem Lectures, 

 Princeton (1912), published in 191 5; Rend. Lincei, April 17, 1910. See also 

 H. Bateman, Cambr. Phil. Trans. 1906. 



2 Phil. Trans. 181 6; Memoirs of the Analytical Society, 18 13 ; Cambr. Phil. 

 Trans. 1820. 



3 Liouville 's Journal and Journal de VEcole Polytechnique, 1832-7; Crelle's 

 Journal, 1834, 11, 1 ; 1834, 12, 273 ; 1835, 13, 219. 



4 Camb. Math.Journ. 1845, 4, 82. 



5 Wcrke (Weber, 1876), 331. 



6 Electrical Papers, 3- 



7 Camb. and Dub I. Math.Journ. 1852, 7, 103. 



