622 SCIENCE PROGRESS 



linear integral equation, which gave to Volterra and Fredholm 

 the basis of their theory of integral equations ; and the theorem 

 discovered independently by Fischer and Riesz is discussed at 

 some length as illustrating most of the modern tendencies of 

 which Van Vleck speaks. Instead of representing a known 

 function by a Fourier series, Fischer and Riesz sought to 

 represent a given Fourier series, or rather its generalisation 

 into a series of orthogonal functions, by a function, under 

 certain precise definitions as to what is meant by the word 

 " represent." Lebesgue integrals were used throughout, and 

 the necessary and sufficient conditions, which were found by 

 Fischer and Riesz, can be expressed very simply. Modern 

 mathematical research is characterised by its tendencies to 

 invert (thus following what is said to have been Jacobi's advice 

 to his pupils) and generalise problems by taking into con- 

 sideration an infinite number of variables, by basing a theory 

 (as Riemann did) on a property rather than on an algorithm, 

 by the presence of existence-theorems, and by the examination 

 of the postulates of pure and applied mathematics. It is very 

 much to the point that Van Vleck emphasises that functions 

 of infinitely many variables appear in important questions of 

 mathematical physics ; thus, the potential due to an electric 

 current in a wire is a function of the shape of the wire, and 

 thus depends upon the non-enumerably infinite number of 

 its points. 



We may here add that even such minute examination of 

 the principles of physics as that of Whitehead (see Science 

 Progress, 191 7, 11, 454) is necessary, for otherwise we cannot 

 tell if our hypotheses are consistent. Thus, if we hold to 

 the hypothesis that no point can belong to more than one 

 body, we have the necessary result that there is no such thing 

 as contact of bodies with surfaces and that, if two bodies are 

 to be in contact, one of them (and only one) must be deprived 

 of a part of its surface. 



G. Andreoli (Giorn. di Mat. 191 5, 53, 97-135) writes on 

 integral and integro-differential equations of a more general 

 type than those considered by Volterra and Fredholm. G. 

 Vivanti (ibid. 209-11) gives an elementary demonstration of 

 a fundamental formula in the theory of integral equations. 



M. Frechet (Bull. Soc. Math, de France, 1915, 43, 248-65), 

 in a paper on the integral of a functional extended to an 



