362 SCIENCE PROGRESS 



vestigates the classification of the substitutions of certain 

 automorphic groups in invariables. 



W. A. Manning, Trans. Amer. Math. Soc, xx (191 9), pp. 

 66-78, continues his researches on the order of primitive groups. 



N. Kryloff, Comptes Rendus, 168 (1919), pp. 721-723, investi- 

 gates approximation formulae based on generalisations of the 

 theory of mechanical quadratures. 



R. L. Moore, Trans. Amer. Math. Soc, xx (1919), pp. 169- 

 178, discusses a set of postulates for plane analysis situs. 



S. Lepschetz, Comptes Rendus, 168 (1919), pp. 672-674, 

 discusses the analysis situs of algebraic varieties in ^-dimen- 

 sional space ; and in a subsequent note (pp. 758-761) he ex- 

 amines Abelian varieties. 



E. Borel, Comptes Rendus, 168 (i9i9),pp. 1 1 78-1 150, discusses 

 the use of decimals to illustrate theorems in the theory of 

 sets, and, in particular, in the theory of sets of zero measure. 



W. H. Young, Proc. Royal Soc, xcvi (a), (191 9), pp. 71-81, 

 makes an important contribution to the theory of the area of 

 surfaces ; after criticising the classical theory and the work 

 of Lebesgue, Peano and Minkowski, the author formulates a 

 theory which appears free from objection ; he gives a defini- 

 tion of the area of a surface which leads to a formula involving 

 a double integral, and proves that the area, so denned, exists 

 if the partial derivates of the co-ordinates with respect to the 

 parametric variables are bounded functions of the parametric 

 variables. 



E. W. Chittenden, Trans. Amer. Math. Soc, xx (19 19), pp. 

 1 79-1 84, continues his researches on the convergence of sequences 

 of functions, ibid., xv (1914), by examining limit functions of 

 sequences which converge ' relatively uniformly.' The same 

 author with A. D. Pitcher, ibid., 213-233, discusses the theory 

 of developments of an abstract class in relation to the Calcul 

 fonctionnel of Frechet. 



G. Fichtenholz, Quarterly Journal, xlviii (191 8), pp. 142-147, 

 discusses conditions under which an integral may be differ- 

 entiated with respect to a parameter under the integral sign. 

 The integrals considered are defined in accordance either 

 with Riemann's or de La Vallee Poussin's or Lebesgue's defini- 

 tion, and conditions for the validity of Leibniz' theorem more 

 general than those previously given by the author and by 

 G. H. Hardy are obtained. 



H. Blumberg, Trans. Amer. Math. Soc, xx (1919), pp. 40-44, 

 extends the concept of ' convex functions ' of a real variable 

 to that of convex functions of n real variables. 



P. Levy, Comptes Rendus, 169 (1919), pp. 375-377, studies 



