368 SCIENCE PROGRESS 



The proof of the existence of the inverse of an analytic 

 function is made to depend, in Weierstrass's theory, on repre- 

 sentation by power series, and, in Cauchy's theory, on the 

 Jacobian of the real and imaginary parts of the function 

 with reference to the real and imaginary parts of the variable. 

 S. Beatty (ibid. 347-53) proves this existence starting from 

 Goursat's conception of an analytic function, and in the method 

 of the proof the theory of sets of points is used. 



Beatty (Amer. Journ. Math. 191 7, 39, 257-62) derives the 

 complementary theorem in the theory of algebraic functions 

 from the Riemann-Roch theorem. In the work of J. C. Fields 

 (191 6) several proofs of the complementary theorem, of which 

 the Riemann-Roch theorem is a particular case, are given. 



K. B. Madhava (Journ. of Indian Math. Soc. 191 7, 9, 141-8) 

 derives some interesting formulae connected with the Zeta 

 function, and, in particular, its addition-theorem and a generali- 

 sation of it. 



W. H. Young (Proc. Roy. Soc. 191 7, A, 93, 42-55) obtains 

 theorems including as particular cases the results on the order 

 of magnitude of the coefficients of a Fourier series obtained by 

 Lebesgue in 19 10. 



In 1 899 the late Willard Gibbs pointed out without proof a 

 hitherto unobserved phenomenon in the behaviour of the 

 approximation curves for a particular Fourier's sine-series at a 

 point of discontinuity, and Gibbs 's work has been developed by 

 Bocher in 1906 and Gronwall in 191 2. H. S. Carslaw (Amer. 

 Journ. Math. 191 7, 39, 185-98) obtains a number of interesting 

 properties of the approximation curves of another particular 

 sine-series by quite simple methods, and shows that all the 

 features of Gibbs 's phenomenon follow immediately from these 

 properties. In conclusion, the extension to the general case of 

 Fourier's series is given, but the method does not differ materi- 

 ally from that of Bocher. 



G. N. Watson (Proc. Lond. Math. Soc. 191 7, 16, 150-74) 

 investigates the theory of Kapteyn series — series in which 

 the terms are multiples of Bessel functions — in which the work 

 of Nielsen (1904), Debye (1909, 1910), and others is taken as 

 starting-point. 



Sir Joseph Larmor (ibid. 8-42), starting from the question 

 as to the possibility of mathematical representation of such 

 irregular graphs as those in the vast collections of meteoro- 



