RECENT ADVANCES IN SCIENCE 525 



Sir Ronald Ross (Science Progress, 1918, 13, 288-98) 

 gives many theories on his " verb-functions," mostly con- 

 nected with " operative involution." 



D. N. Lehmer (Amer. Journ. Math. 191 8, 40, 375-9°) 

 gives an arithmetical theory of certain Hurwitzian continued 

 fractions, of which the origin was the empirical discovery of 

 such facts as that the denominator of the convergent of order 

 7,n in the regular continued fraction which represents e is 

 divisible by n. The theory of continued fractions is also 

 treated by I. Schur (12), O. Szasz (54), G. Humbert (32, 33), 

 A. Arwin (67), and A. Pringsheim (27). Questions about 

 infinite series in general are investigated by F. Carlson (11), 

 A. Pringsheim (26), M. Petrovitch (32), and W. Gross (62, first 

 paper) ; and special series (product for trigonometric functions) 

 by G. Kowalewski (18), and (Tschebychef's polynomials) by 

 J. Chokhate (35). 



Analysis. — On the theory of functions of real variables — 

 continuity and differentiation : H. Hahn (17, 57), K. Knopp 

 (17), C. Burstin(6o,6i), A. Haar(23),Ch. J. de la Vallee Poussin 

 (39), and C. Caratheodory's (68) book of 191 8 ; integration : 

 G. A. Bliss (8), S. Lefschetz (46), O. Perron (27), F. Riesz (17, 

 55), and A. Denjoy(3i). On trigonometric series, see U. Dini 

 (46), M. Angelesco (32), G. H. Hardy and J. E. Littlewood (34), 

 W. H. Young (32), M. Akimoff (34), G. Szego (53-4), and 

 L. Fejer (61) ; and on orthogonal functions G. D. Birkhoff (8), 

 W. Sternberg (8), and H. Laudien (19). 



The general theory of analytic functions of complex vari- 

 ables is dealt with by R. Konig (13, 17), J. von S. Nagy (15), 

 L. Fejer (15-16, 26-7), W. Gross (17), J. Schur (18), G. Hamel 

 (22), G. Pick (23), R. Jentsch (23), G. P61ya (23), L. Bieberbach 

 (23), G. Remoundos (32), M. Frechet (33), H. von Koch (65, 

 67), F. R. Berwald (65-6), S. Wigert (65), A. Denjoy (35, 48, 

 49), F. Iversen (35), D. Pompeiu (36), G. Valiroi\(4i), M. Beke 

 (53), F. Riesz (54), and P. Csillag (55) ; and conformal repre- 

 sentation by G. Pick (58), R. Furth (56), and E. Hille (64). 



For special functions, we have (Gamma functions, factorials, 

 etc.), N. Nielsen (n), P. H. Dojes (49-50), E. Stridsberg (64) 

 arithmetical functions, E. Hecke (14) I Legendre's functions, 

 A. Haar (21), W. H. Young (33), P. Humbert (34) ; Dirichlet's 

 series, E. Cotton (40) and M. Fekete (54). A. B. Coble (Amer. 

 Journ. Math. 191 8, 40, 317-40) showed that the connection 



