124 



SCIENCE 



[N. S. Vol. XXXIX. No. 995 



in the totality of functions mathematically 

 conceivahle. 



Even for functions of Class or 1 the 

 trigonometric series has a limited power of 

 representation. This is manifest from an 

 example given by Paul Du Bois Reymond of 

 a continuous function which can not be rep- 

 resented by a trigonometric series. It re- 

 mains to determine in the future just what 

 properties are necessary and sufficient to 

 characterize those functions of Classes 

 and 1 which are expressible by means of 

 trigonometric series. 



Earlier in my paper I pointed out that 

 the generality of functions representable 

 through Fourier's series was so great that 

 the mathematician was led irresistibly to 

 the Dirichlet definition of a function. If, 

 namely, to every value of x in an interval 

 we have a corresponding value of y, then 

 y is called a function of x, no matter how 

 the correspondence is set up, whether by a 

 graph, a mathematical expression, a law, or 

 any other way. To-day the pendulum has 

 swung back to the old question of Euler. 

 The study of representability in terms of 

 trigonometric series has been succeeded by 

 the broader question of the possibility of 

 analytic expression in general. Now every 

 continuous function, as is well known, can 

 be represented by a uniformly convergent 

 set of polynomials. Starting then from the 

 totality of polynomials as a basis of func- 

 tions for Class 0, we arrive successively at 

 Baire's and Lebesgue's classes of func- 

 tions corresponding to or, if you prefer, 

 marked, by the transfinite numbers of the 

 first and second classes. 



Do these different classes of functions 

 comprise all which a.ve" analytically expres- 

 sible"? Before answering the question it 

 is necessary first to sharply define the 

 phrase ' ' analytically expressible. ' ' This is 

 done by Lebesgue. Then, after broaden- 

 ing the content of these classes in a manner 



I have not the time to describe, he goes on 

 to show that they do in truth comprise all 

 such functions. The final question then 

 confronts us : Are all possible functions in- 

 cluded which are defined in accordance with 

 the general definition of Dirichlet? In 

 other words, are there functions incapable 

 of being "analytically expressed"? Le- 

 besgue by an example shows that this is the 

 case. Our study of the Fourier series 

 opened with the question: What is an ar- 

 bitrary function? Here, at last, appar- 

 ently, we have discovered the existence of a 

 function of such a height or depth of ar- 

 bitrariness as to be mathematically inex- 

 pressible. Having started with the Fou- 

 rier series on a voyage of exploration, shall 

 we conclude by saying that there is for us 

 an unattaina;ble pole? 



Edwaed B. Van Vleck 

 University op Wisconsin 



UNIVEBSITT REGISTRATION STATISTICS 

 The registration returns for November 

 1, 1913, of thirty of the leading universities 

 of the country will be found tabulated on 

 the following page. Specific attention 

 should be called once again to the fact that 

 these universities are neither the thirty lar- 

 gest universities in the country, nor neces- 

 sarily the leading institutions. The only 

 universities which show a decrease in the 

 grand total attendance (including the sum- 

 mer sessions) are Harvard, Western Re- 

 serve and Yale, the attendance of the two 

 institutions last named having remained 

 practically stationary. The largest gains 

 in terms of student units, including the 

 summer attendance, but making due al- 

 lowance by deduction for the summer 

 session students who returned for instruc- 

 tion in the fall, were registered by New 

 York University (965), Illinois (944), Co- 

 lumbia (927), Wisconsin (749), Pennsyl- 

 vania (681), California (614), Iowa (598), 



