8 SCIENCE PROGRESS 



According to the Rev. sent. (1917, 25 [2], 46) B. Hostinsky 

 (Casopis pro pest. math, a fys. Prague, 191 5, 44, 28-30) finds 

 that the function determined by Cauchy's theorem is the 

 geometrical mean of its values at the points of the given contour. 



W. Sierpinski (Rend. del. Circ. mat. di Palermo, 1916, 41, 

 187-90) has a note on a potential series which is convergent at 

 every point of its circle of convergence and represents on this 

 circle a discontinuous function. 



A. Signorini (Ann. di Mat. 191 6, 25, 253-74) writes on the 

 conditions under which a function exists which is regular in 

 a simply connected region S of the complex plane and of 

 which the real and imaginary parts reduce, on certain arcs of 

 the contour of S, to functions of the arc of this contour. 



M. Kossler (Rozp. CeskS Akad. Prague, 191 5, 24, No. 41), 

 starting from the integral of Cauchy, studies developments 

 in series of analytic functions which hold in a given domain. 



J. L. W. V. Jensen (Mem. de I' Acad. Roy. de Danemark, 

 Copenhagen, 191 6, 2, 200-28) investigates a class of funda- 

 mental inequalities for the absolute amounts of analytic func- 

 tions under certain conditions. There is a historical sketch 

 and an account of the new researches of the author. 



H. Bohr (Nyt Tidsskr. for Mat. 191 6, 27, 73-8) proves an 

 inequality concerning the manner of increase of the values of 

 analytic functions. 



M. Beke (Math, es term. ert. Budapest, 191 6, 34, 1-61) 

 proves and applies the theorems of Hadamard and Hurwitz on 

 certain series compounded out of two given power series. 



E. Borel's important Lecons sur les Fonctions Monogenes 

 is reviewed at length in the present number of Science Pro- 

 gress ; it bears a close relation to the two papers by Borel 

 noticed in the last number of Science Progress (191 8, 12, 544 ; 

 cf. the review of the Book of the Rice Institute in the present 

 number). 



On Dirichlet's series, we may refer to P. Nalli (Rend, del 

 Circ. mat. di Palermo, 191 7, 42, 61-72), and A. Arwin (Mem. 

 de I' Acad. Roy. de Danemark, Copenhagen, 191 6, 2, 79-85), 

 and, on the Zeta-f unction, to C. de la Vallee Poussin (Compt. 

 Rend. 1 91 6, 163, 418-21, 471-3). 



R. D. Carmichael (Amer. Journ. Math. 191 7, 39, 385-403) 

 gives a sequel to his memoir of 191 6 (Science Progress, 191 7, 

 11, 456-7 ; cf. 1918, 12, 370) in which he laid the foundations 



