176 SCIENCE PROGRESS 



T. Varopoulos {C.R., 174, 1922, 89, 1323) and S. Saranto- 

 poulos {ibid., 1320) study increasing functions ; and R. 

 Nevanlinna {ibid., 1325) the relations between the order of growth 

 of a function and the density of its zeros. 



G. Vahron {C.R., 174, 1922, 1054) has a note on integral 

 functions of integral order. 



Dirichlet, in his theory of Fourier series, did not, of course, 

 consider uniform convergence, and the more recent theories, 

 of Heine, Cantor and Du Bois Reymond, proceed on different 

 lines ; C. Neumann {Leipzig Ber., 73, 1921, 201-14) modifies 

 Dirichlet's theory to fill the gap. 



N. Abramescu {Rend. Lincei, 31, 1922, 89, 152, 197) deals 

 with the series of polynomials known as Darboux series. 



G. Szego {Math. Ann., 82, 1921, 188-212 ; Math. Zs., 

 12, 1922, 61-94) examines the convergence of orthogonal 

 developments of functions ; S. Banach {Proc. L.M.S., 21, 

 1922, 95-7) gives an example of such a development whose 

 sum is everywhere different from the developed function, 



T. Carleman {C.R., 174, 1922, 1680) investigates a non- 

 decreasing function ^(a:) satisfying the relations 



x''dj>{x) =C„ {v =0, I, 2 . . .), 



J — CD 



where Cq, Cj ... is a given sequence of real constants ; this 

 is a generalisation of Stieltjes' problem of moments. 



L. Fejer {Math. Ann., 85, 1922, 41-8) investigates the 

 zeros of polynomials which arise from minimum conditions, 

 e.g. the Tschebychef and Legendre polynomials ; A. Angelesco 

 {C.R., 174, 1922, 273) generalises the theory of the zeros of 

 orthogonal polynomials, and G. Szego {Math. Ann., 86, 1922, 

 1 14-39) examines their asymptotic expressions. 



A. Angelesco {Rend. Lincei, 31, 1922, 236-9) also writes 

 on the Laguerre polynomials. 



E. L. Ince {Proc. Camb. Phil. S., 21, 1922, 117-20) shows 

 that the second solutions of Mathieu's equation cannot be 

 periodic. 



P. Humbert {C.R., 174, 1922, 91) examines solutions of 

 Laplace's equation in four variables applicable to hyper- 

 cylinders having for base hyperboloids of revolution ; he thus 

 obtains Mathieu functions in two variables. 



P. J. Myrberg {C.R., 174, 1922, 1402) examines the essential 

 singularities of automorphic functions of several variables. 



G. Mittag-Leffler {C.R., 174, 1922, 789) and E. Goursat 

 {ibid., 836) continue their priority dispute concerning the 

 proof of Cauchy's theorem. 



N. E. Norlund {C.R., 174, 1922, 919) has a note on the con- 

 vergence of Stirling's interpolation series. 



