RECENT ADVANCES IN SCIENCE 623 



abstract aggregate, gives a generalisation of de la Vallee- 

 Poussin and Lebesgue's definitions of an integral. 



Geometry. — G. Ascoli {Giorn. di Mat. 191 5, 53, 203-8) 

 gives a non-Archimedean construction of plane homography. 



L. Bieberbach (Jahresber. der Deutschen Math.-Ver. 191 5, 

 24, 247-50) proves rigorously the famous old theorem that 

 among all plane finite domains of a given breadth the circle 

 has the greatest content. Cf. the same author's paper in the 

 Math. Ann. (191 5, 77, 153-72). 



H. R. Kingston (Amer. Journ. Math. 191 6, 38, 407-30) 

 lays the foundation for the study of the metric differential 

 properties of nets of plane curves, Wilczynski having discussed 

 their projective differential properties in 191 1. 



H. C. Gossard {ibid. 431-45) discusses whether there are 

 lines connected with a tetrahedron such that, if the vertices 

 are reflected in these lines, the reflections will fall on the 

 opposite faces ; and, if so, what the locus of these lines is. 



In 191 5 C. H. Sisam discussed sextic surfaces whose plane 

 sections are of genus 1. In Amer. Journ. Math. (1916, 38, 

 373-86) he considers non-ruled sextics whose plane sections 

 are of genus 2 . 



L. P. Eisenhart {Trans. Amer. Math. Soc. 1916, 17, 437- 

 58) determines certain deformable transformations of Ribau^ 

 cour. The subject is in close connection with some of 

 Bianchi's work in his Differenziale geometria and also in papers 

 published by him in 1906 and 191 5, and with some previous 

 (191 5) results of the author. 



ASTRONOMY. By H. Spencer Jones, M.A., B.Sc, Royal Observatory, 

 Greenwich. 



Einstein's Theory of Gravitation. — In the Physics section of 

 these notes in the last number of Science Progress (p. 461), 

 Einstein's new theory of gravitation was referred to. Inasmuch 

 as the only evidence which can be adduced either in favour 

 of or against the theory must be derived from astronomical 

 observations, the astronomical aspect of the question is of some 

 considerable importance. The evidence at present available 

 and the directions in which further evidence may be obtained 

 are accordingly summarised below. The metaphysical aspect 

 of the theory and the nature of the new law of gravitation are 

 j not considered here, as limitations of space would not permit. 



