RECENT ADVANCES IN SCIENCE 545 



W. H. Young (Proc. Lond. Math. Soc. 191 7, 16, 273-93) 

 proves the general theorem of integration by parts in a form 

 which takes account of recent progress in the theory of integ- 

 ration, and deduces from it the second theorem of the mean 

 in an extended form and for any number of variables . 



G. Polya (Acta Math. 191 7, 41, 99-118) investigates those 

 power-series whose circles of convergence are natural limits. 

 The subject has been hitherto made obscure by the too frequent 

 use of the phrase that a power-series cannot " in general " be 

 continued beyond its circle of convergence : of course both 

 continuable and non-continuable power-series have the same 

 cardinal number. P61ya considers a functional space of 

 infinitely many dimensions whose points are the power-series 

 converging in the unit circle, and proves the theorem : the 

 aggregate of non-continuable power-series has only inner points 

 and is everywhere dense ; while the aggregate of continuable 

 power-series is nowhere dense and perfect. 



G. H. Hardy (Proc. Camb. Phil. Soc. 191 7, 19, 60-33) general- 

 ises a theorem published by Polya in 191 5, on transcendental 

 whole functions which assume integral values for all positive 

 integral values of the variable, by a slight modification of 

 Polya's own argument, without the addition of any essentially 

 new idea to those which he employed. 



A. Wiman (Acta Math. 1916, 41, 1-28) investigates the con- 

 nection between the maximum M(r) of the modulus of an 

 analytic function and the maximum of the real part, for a 

 given argument of the function, and very greatly generalises a 

 result of Borel (1897) f° r whole functions. Wiman shows 

 that light is thrown by it both on the famous theorem of Picard 

 and on the properties of the solutions of differential equations 

 in the neighbourhood of points of indefiniteness. 



K. B. Madhava (Jonrn. Indian Math. Soc. 191 7, 9, 186-201) 

 gives an account, with many references, of asymptotic expan- 

 sions of integral functions. 



G. N. Watson (Proc. Camb. Phil. Soc. 191 7, 19, 42-8) proves 

 rigidly the formulae of approximations for (1) a Bessel function 

 of equal and large order and argument, and (2) its first deriva- 

 tive, which are due to Cauchy (1854), Graf and Gubler (1898), 

 Debye ( 1 909), and others, without the use of contour integration, 

 on the one hand, and without appealing to physical arguments 

 such as Kelvin's (1887) " principle of stationary phase," on 



