RECENT ADVANCES IN SCIENCE 181 



R. Jentzsch (Acta Math. 191 7, 41, 219-51) proves the 

 theorem, which gives a more general case of some results of 

 Hurwitz, Lindwart and Polya, and Lukacs, that every point 

 of the circle of convergence of a power-series is a point of 

 condensation of the zero of the " segments " (sums of the first 

 n terms of the series, where n is any finite number). This new 

 theorem is also related to the general researches of Montel, 

 Severini, Caratheodory, and Landau on sequences of one- 

 valued analytic functions. Jentzsch (ibid. 253-70) continues 

 these researches. The memoir first named also appeared at 

 Berlin in 1 914 as a dissertation. 



J. Hodgkinson (Proc. Lond. Math. Soc. 191 8, 17, 17-24) 

 applies the method of conformal representation given by 

 Burnside in 1893 to the discussion of certain results obtained by 

 Gauss and Kummer with respect to the hypergeometric series. 



P. Humbert (Proc. Roy. Soc. Edinburgh, 191 8, 38, 61-9) 

 finds a reduction formula for the functions of the second kind 

 connected with the polynomials of applied mathematics. 



A. Ostrowski (Acta Math. 191 7, 41, 271-84) investigates 

 solutions of the functional equation (f>(x) . <f)(y) = <$>(xy), where 

 the function only takes real values, the arguments take the 

 values of any assigned corpus, and 4>(x -f- y) ^ <j>(x) + 4>(y)- 



F. W. Reed (Amer. Journ. Math. 191 8, 40, 97-107) applies 

 the method of infinitesimal transformations developed by Lie 

 to those types of integral invariants defined and discussed by 

 Poincare (1890, 1895), Lie (1897), an d others. 



R. D. Carmichael (Annals of Math. 191 8, 19, 159-71), 

 starting from Birkhoff's (1908) work on adjoint systems of 

 boundary conditions associated with adjoint linear differ- 

 ential equations of order n, obtain certain extensions of one 

 of C. Sturm's (1836) theorems of comparison to equations of 

 order higher than the second. 



D. Buchanan (Proc. Lond. Math. Soc. 191 8, 17, 54-74), in 

 connection with his work of 191 5, determines solutions of the 

 differential equations of motion which give orbits asymptotic 

 to an " isosceles-triangle solution of the problem of three 

 bodies." 



Geometry. — J. Rey Pastor (Scientia, 191 8, 23, 413-22) gives 

 a slight account of the systematisation of geometry by the 

 theory of groups, which was brought about chiefly by Klein's 

 11 Erlangen Programme " of 1872. 



