RECENT ADVANCES IN SCIENCE 363 



the concept, due to Ra Gateaux, of mean value in a functional 

 domain. 



W. H. Young, Proc. Royal Soc, xcvi (a), (1919), pp. 82-91, 

 investigates formulae for the change of independent variables 

 in a multiple integral. 



A. Buhl, Comptes Rendus, 168 (19 19), pp. 504-506, discusses 

 the interchange of parameter and argument of certain alge- 

 braic integrals. 



P. Faton, Comptes Rendus, 168 (1919), pp. 501-502, criticises 

 a recent note by A. Denjoy (ibid., p. 387) on lines of singularities 

 of analytic functions, pointing out that Denjoy's results are 

 but slight generalisations of previously known theorems. 

 Denjoy subsequently (ibid., pp. 848-851) applies his results to 

 the evaluation of definite integrals. 



Carleman, Comptes Rendus, 168 (1919), pp. 843-845, gives an 

 elementary proof of a theorem proved by Koeble in 1906, 

 that ' Every function establishing a conformal representation 

 between two domains, each bounded by a finite number of 

 circles (one of which includes all the others which are mutually 

 exclusive) is a linear function.' An extension is made to the 

 case in which the boundaries are ' simple curves.' 



A. Denjoy, Comptes Rendus, 169 (191 9), pp. 219-221, 

 examines the connexion between Riemann integration and 

 Lebesgue integration. 



G. H. Hardy and J. E. Littlewood, Proc. London Math. 

 Soc. (2), xviii (1919), pp. 205-235, discuss the most far-reaching 

 generalisation of Tauber's theorem, the converse of Abel's well- 

 known theorem concerning the continuity of a power series 

 along a radius up to its circle of convergence. 



G. H. Hardy, Messenger, xlviii (191 8), pp. 107-1 12, gives the 

 first elementary proof of a theorem due to Hilbert that, when 

 a n is positive, the convergence of the series Ha H 2 entails the 

 convergence of the double series %%a m a n (m + n). 



J. W. Hopkins, Trans. Amer. Math. Soc, xx (1919), pp. 

 245-259, investigates convergent developments associated with 

 irregular boundary conditions ; the function satisfying the 

 assigned boundary conditions is a solution of a certain linear 

 differential equation of the third order, and the analysis is 

 based on Birkhoff's discussion, ibid., ix (1908). 



G. Remoundos, Comptes Rendus, 168 (1919), pp. 1265-1268, 

 discusses the summability of asymptotic solutions of linear 

 differential equations of the first order, and establishes results 

 left unproved by Borel in his well-known Lecons stir les series 

 diver gentes. 



G. Julia, Comptes Rendus, 168 (1919), pp. 502-504, states a 

 general property of integral functions which is associated 

 with Picard's theorem, that the function assumes all finite values 



