RECENT ADVANCES IN SCIENCE 351 



On conformal representation we may refer to P. Koebe 

 (Journ. fur Math. 147, 67-104), G. Faber (Sb. Miinchen, 1916, 

 39-42), and P. Montel (Compt. rend. 191 7, 164, 879-81). 



F. Schottky (Journ. fur Math. 147, 161-73) gives a con- 

 spectus of the elementary considerations which occur in proofs 

 of Picard's famous theorem. 



On the general theory of functions we may also mention the 

 papers of R. Konig (Math. Ann. 191 7, 78, 63-93) on a theory 

 of Riemann's pairs of functions, P. Montel (Ann. de Vic. norm. 

 [3], 33, 223-302) on certain " normal " families of analytic 

 functions, and G. Polya (Vjsschr. Zurich, 1916, 61, 531-45) on 

 the rapidity of convergence of a power-series which represents 

 a whole transcendental function satisfying an algebraic differ- 

 ential equation. 



R. D. Carmichael (Amer. Journ. Math. 191 8, 40, 113-27) 

 continues his wo*rk of 191 6 and 191 7 (Science Progress, 191 8, 

 13, 8-9) on functions defined by certain series, and makes a 

 contribution towards solving the problem of representing given 

 functions in the form of series Sc n g(x + n)/g(%)> H. Gronwall 

 (Ann. de I'ec. norm. [3], 33, 301-93) finds the zeros of the func- 

 tions P(z) and Q(z) associated with the Gamma function. On 

 Hyperfuchsian and Hyperabelian groups and functions and 

 certain total differentials, see E. Picard (ibid. 363-72, 373-9) 

 and G. Giraud (ibid. 303-29, 330-62 ; Compt. rend. 191 7, 164, 

 386-9, 487-9). On conditions for developability in Dirichlet's 

 series, see J. F. Steffensen (Nyt Tidsskr. for Mat. 191 7, 28, 9-1 1). 

 On two points in the theory of trigonometric series, see I. 

 Priwaloff (Compt. rend. 1917,165, 96-9) and H. Hahn (Jahresber. 

 der D.M.V. 191 6, 25, 359-66). 



G. N. Watson (Proc. Roy. Soc. 191 8, 94a, 190-206 ; Nature, 

 191 7, 100, 299) gives some general theorems concerning the 

 zeros of Bessel functions : the theorems are true for functions 

 of any order, and, unlike results previously known, are of 

 particular interest in the case of functions of high order. Watson 

 (Proc. Lond. Math. Soc. 191 8, 17, 116-48) publishes a second 

 part to his paper of 19 10 on the harmonic functions associated 

 with the parabolic cylinder. In this paper, which is much 

 more closely connected with a paper of 191 7 (cf. Science 

 Progress, 191 8, 12, 368), he investigates various types of 

 asymptotic formulae and expansions which are to be associated 

 with those previously obtained for Bessel functions. 



