196 SCIENCE PROGRESS 



51, 137-41) writes on the theorems of the mean and of 

 Rolle. 



W. H. Young (Compt. Rend. 1916, 162, 909-12) has a note 

 on the foundations of the theory of integration, and N. Lusin 

 (ibid. 975-8) has one on the finding of primitive functions, 

 giving the results of his Russian thesis on integrals and trigono- 

 metric series. 



W. H. Young (ibid. 163, 187-90) proves a theorem on the 

 convergence of Fourier's series. 



T. W. Chaundy (Mess, of Math. 45, 1 1 5-9) gives a condition 

 for the validity of Taylor's expansion. 



G. Mittag-Leffler (Sitzungsber. der k. b. Akad. der Wiss. zu 

 Munchen, 1915, 419-24) gives a new and simple proof of the 

 theorem of Serge Bernstein on the necessary and sufficient 

 conditions that a function of a real variable should be analytic 

 along a certain straight line. G. Fubini (Atti della R. Accad. 

 delle Scienze di Torino, 191 5-6, 51, 538-40) also has a note on the 

 theorems of Bernstein and Pringsheim on development in 

 Taylor's series. M. Riesz (Acta Math. 40, 337-47) proves 

 Bernstein's theorem by means of Lagrange's interpolation- 

 formula ; and also gives (ibid. 349-61) a theorem of convergence 

 for Dirichlet's series. 



F. Schottky (Journ.fur Math. 146, 234-44) gives a historical 

 sketch of the work of Green, Cauchy, Riemann, Goursat, and 

 Moore on Cauchy's integral, and also simplifications and 

 generalisations of Moore's method. 



Short notes on Poisson's integral considered as a direct 

 consequence of Cauchy's integral were given by V. Laska 

 (Casopis pro pestovdni mathematiky a fysiky, Prague, 191 3, 42, 

 398-401) and K. Petr (ibid. 556-8). 



J. L. Walsh (Annals of Math. 1916, 18, 79-80) establishes 

 Cauchy's integral-formula by means of the theorem called " the 

 mean value theorem for harmonic (or conjugate) functions." 

 This process is the reverse of the one used in, for example, 

 Burkhardt's Funktionentheorie by the use of essentially the 

 same analytic machinery. The proof is one given by M. 

 Bocher (Bull. Amer. Math. Soc. 1894-5, 1» 206-7) i n a P a per 

 on Gauss's third proof of the fundamental theorem of algebra. 



G. Kowalewski (Sitzungsber. der k. Akad. der Wiss. in Wien, 

 191 5, 124 [11a], 333-8) develops at length the proof of the 

 existence of implicit functions which was indicated in his book 



