RECENT ADVANCES IN SCIENCE 455 



in German) and Moritz Pasch's important paper of 1887 will be 

 remembered in this connection. Lately, Pasch (Monatshefte 

 fiir Math, and Physik, 191 5, 26, 303) has returned to and further 

 illustrated the subject. 



L. L. Silverman (Trans. Amer. Math. Soc. 1916, 17, 284) 

 discusses some points in the extension, first carried out by Paul 

 du Bois-Reymond in 1887, of Cesaro's and Holder's definitions 

 of summability of series to the case of functions of a continuous 

 variable. 



In 1914 C. A. Fischer gave a definition of the derivative of a 

 function of a surface which is analogous to Volterre's definition 

 of the derivative of a function of a line, and proved that if the 

 derivative is continuous and approached uniformly, the first 

 variation of the function is equal to the double integral of the 

 derivative multiplied by the first variation of the dependent 

 variable. Fischer now (Amer. Journ. Math. 1916, 38, 259) 

 extends the theory to the case where there are points or curves 

 where the derivative does not exist, and applications are made 

 to problems of the calculus of variations. 



G. H. Hardy (Trans. Amer. Math. Soc. 1916, 17, 301) gives 

 a new method, which is less elementary but considerably more 

 powerful than those used hitherto, for the discussion of Weier- 

 strass's continuous but not differentiable function and similar 

 questions. Several very general results are proved concerning 

 Weierstrass's function and the corresponding function defined 

 by a series of sines, and towards the end of the paper are : (1) a 

 simple example of a function represented by an absolutely con- 

 vergent Fourier series which does not satisfy a " Lipschitz 

 condition " of any order for any value of x ; (2) a proof of S. 

 Bernstein's theorem (1914) that if f(x) satisfies a Lipschitz 

 condition of order greater than one-half throughout an interval, 

 its Fourier series is absolutely convergent, and one-half is 

 the least number which possesses this property ; (3) a dis- 

 cussion of the function which, according to Paul du Bois- 

 Reymond (1875), was stated by Riemann to his pupils to have 

 no finite differential quotient for any one of an everywhere- 

 dense set of values of x : Hardy proves Riemann 's assertion 

 and a good deal more. 



An example of the need which mathematicians now feel 

 for proving " obvious " things is the demonstration, by E. 

 Sciolette (Giorn. di Mat. 1914, 52, 84), that if f(x) is positive 



