366 SCIENCE PROGRESS 



ciates an advance on the results already known concerning the 

 situation of the zeros of the zeta-function of Riemann, and 

 states that it can be proved without employing the celebrated 

 " Riemann hypothesis." Subsequently {ibid., pp. 11 53-1 154) 

 he proves a theorem, concerning limits, which is of importance 

 in connexion with the distribution of prime numbers. 



G. H. Hardy, Proc. London Math. Soc. (2), xviii (1919), pp. 

 201-204, gives a short supplement to his paper on " The Aver- 

 age Order of the Arithmetical Functions P(n) and A («)," ibid., 

 (2), xv (191 6), pp. 192-213 ; these functions are two functions 

 which make their appearance in connexion with the investiga- 

 tion of the number of representations of n as the sum of two 

 squares and with the number of divisors of n. 



G. H. Hardy and J. E. Littlewood, Quarterly Journal, xlviii 

 (1919), pp. 272 et seq., give a new solution of Waring's problem ; 

 the assertion made by Waring was that every positive integer 

 is expressible as the sum of at most four squares, nine cubes, 

 nineteen fourth powers, and so on. The authors give a proof, 

 based on the use of Cauchy's theorem, which has no points of 

 contact with the previously known solution, due to D. Hilbert 

 Math. Ann., lxvii (1909), and subsequently simplified by his 

 students. 



A. J. Pell, Trans. Amer. Math. Soc, xx (1919), pp. 23-39, 

 discusses linear equations in infinitely many unknowns with 

 unsymmetric coefficients. Such equations include linear in- 

 tegral equations with unsymmetric nuclei as a special case ; 

 the present paper contains extensions of the theorems con- 

 cerning integral equations which have previously been dis- 

 cussed by the author in 1910-11. 



J. Drach, Comptes Rendus, 168 (191 9), 497-501, discusses 

 the integration by quadratures of a class of differential equa- 

 tions of the second order. 



P. Boutroux, Comptes Rendus, 168 (1919), pp. 11 50-1 152, 

 discusses definitions of multiform functions, with a view to 

 obtaining representations of the totality of their branches 

 throughout their domain of existence. Later (ibid., pp. 1307- 

 13 10) he discusses a particular class of such functions which are 

 integrals of a differential equation of the first order. 



H. T. H. Piaggio, Phil. Mag. (6), xxxvii (1919), pp. 596- 

 600, makes a contribution to the numerical integration of 

 differential equations of the first order ; his results are an 

 extension of those due to Runge, Math. Ann., xlvi (1895), and 

 in certain cases are more accurate than approximations pre- 

 viously given. 



R. Gamier, Comptes Rendus, 169 ( 1919), pp. 223-225, states 



