GENERAL PAPERS 43 



We may go still further, however. Many laws of science 

 are the solutions of either total or partial differential equa- 

 tions. Now it is generally supposed that if we have the second 

 derivative of a variable given in an equation, we may from 

 this equation find the third derivative, then the fourth, etc., 

 and thus for a domain not too large find the coefficients of a 

 Taylor series for the function. That is to say more briefly 

 we find that most functions are assumed to be analytic. Now 

 it is well known that in a wide field of physics these analytic 

 solutions are, even if possible, of no interest, and that we must 

 seek for solutions which are at least nonanalytic on certain 

 boundaries. But we can still advance owing to the investi- 

 gations of Borel, for he invented a differentiable function 

 which in no region, however small, is analytic. This function, 

 for example, would permit lis to study the potential in a region 

 which was discontinuous at a set of points everywhere as dense 

 as rational numbers. We do not then have to assume that if 

 a function is continuous, and it has derivatives that it is 

 necessarily analytic. 



Turning now to another class of investigations, we find that 

 it is generally assumed in some sciences that the succession of 

 phenomena depend only upon those phenomena that immedi- 

 ately precede. Indeed, this condition permits the use of cal- 

 culus. But the assumption is purely gratuitous, for it is pos- 

 sible to devise functions in which not only the preceding state 

 but other preceding states act upon the present. It is as if the 

 remote past can reach out a ghostly hand and affect the pres- 

 ent. The bearing such a possibility has upon heredity is evi- 

 dent at once. And the mathematician, since Volterra devised 

 such functions, can assert that the assumption of no action 

 over an interval of time is simply an assumption. 



A still different kind of function is also due to Volterra and 

 others, namely the function dependent upon a whole infinity of 

 independent variables. Usually the number of independent 

 variables is assumed to be relatively small. The whole ten- 

 dency of science is to reduce the number of causes. This may 

 be due to the fact that functions of an infinity of variables had 

 never been studied. But this obstacle is now removed and in 

 the study of functions of lines, surfaces, etc., as well as various 

 functional spaces, we have the development of a means of 



