jANtlAEY 23, 1914] 



SCIENCE 



123 



siderations, the wonderful pliability of the 

 series. 



It was this pliability which was embodied 

 in Fourier 's intuition, commonly but falsely 

 called a theorem, according to which the 

 trigonometric series (I.) "can express any 

 function whatever between definite values 

 of the variable." This familiar statement 

 of Fourier's "theorem," taken from 

 Thompson and Tait's "Natural Philos- 

 ophy," is much too broad a one, but even 

 with the limitations which must to-day be 

 imposed upon the conclusion, its impor- 

 tance can still be most fittingly described 

 as follows in their own words : The theorem 

 "is not only one of the most beautiful re- 

 sults of modern analysis, but may be said to 

 furnish an indispensable instrument in the 

 treatment of nearly recondite question in 

 modern physics. To mention only sono- 

 rous vibrations, the propagation of electric 

 signals along a telegraph wire, and the 

 conduction of heat by the earth's crust, as 

 subjects in theif generality intractable 

 without it, is to give but a feeble idea of 

 its importance." 



Truly, the theorem is so comprehensive 

 in its mathematical content that we mathe- 

 maticians may well query with one of my 

 colleagues whether it may not have con- 

 ditioned the form of physical thought it- 

 self — whether it has not actually forced the 

 physicist often to think of complicated phys- 

 ical phenomena as made up of oscillatory 

 or harmonic components, when they are not 

 inherently so composed. 



It is this same pliability of the series that 

 has been a source of perpetual delight and 

 surprise to the mathematician. It has re- 

 vealed an undreamt-of power in analysis. 

 It has stimulated intuition and vigor, and 

 has helped to usher in a modern critical 

 era in mathematics similar in spirit to the 

 Greek period. It has separated differen- 

 tiable from continuous functions; it has 



put the integral calculus on a basis of in- 

 dependence of the differential calculus; it 

 has focused attention upon sets of irregu- 

 larities and discontinuities whose study has 

 started the point-set theory; it has opened 

 the field of discontinuous functions to anal- 

 ysis and, above all, has engendered a theory 

 of functions of the real variable. 



To the mathematician the theory of ana- 

 lytic functions for some time appeared to 

 be of much greater importance than the 

 freaky theory of the real variable, because 

 almost all the important functions of mathe- 

 matics are analytic. Also, the same has 

 been hastily assumed for physics because 

 the real and imaginary components of an 

 analytic function are harmonic functions 

 satisfying Laplace's equation. But this is 

 to ignore features of at least equal, if not 

 of superior, importance. Not long ago 

 many thought that the mathematical world 

 was created out of analytic functions. It 

 was the Fourier series which disclosed a 

 terra incognita in a second hemisphere. 



Here, in the new hemisphere, the mathe- 

 matician has advanced beyond the boun- 

 dary of the trigonometric series. It has been 

 found that discontinuous functions repre- 

 sentable through such series form a thor- 

 oughly restricted class. They belong to 

 what Baire calls the first class of functions 

 which are limits of convergent sequences 

 or series of continuous functions, them- 

 selves of ' ' class 0. ' ' These in turn may be 

 used to generate new functions. Even as 

 non-uniformly convergent Fourier series 

 may give rise to discontinuous functions of 

 Class 1, so non-uniformly convergent series 

 of functions of this class may give a new 

 sort of functions of Class 2, and so on. 

 Indeed, to every transfinite number a of the 

 first or second class there corresponds, as 

 Lebesgue has shown, a definite class of func- 

 tions. Thus the Fourier series has, after 

 all, a very limited range of representation 



