RECENT ADVANCES IN SCIENCE 457 



have certain types of singularities at infinity. These re- 

 searches are connected with some work of Poincare (1885, 1886), 

 Norlund (1914), and the theory of Dirichlet's series. 



H. Bohr (Jahresber. der Deutsch. Math. Ver. 191 5, 24, 1) 

 gives a report on the present state of the theory of Riemann's 

 Zeta-f unction from the point of view of the theory of functions. 

 E. Landau (Math. Ann. 191 5, 76, 212) gives some researches 

 relating to Hardy's discovery of an infinity of zeros of the 

 Zeta-function with the real part \, and obtains a great sim- 

 plification of Hardy's proof. 



On the theory of Riemann's Zeta-function we may also refer 

 to a paper by H. Bohr (Acta Math. 40, 67). 



Dr. C. R. Dines (Proc. Land. Math. Soc. 1916, 15, 243) dis- 

 cusses points arising from Prof. E. H. Moore's paper of 191 2 

 on the foundations of the theory of linear integral equations 

 and which are connected with Moore's important and well- 

 known " general analysis " of 1906. 



Geometry. — E. M. Langley (Math. Gaz. 1916, 8, 268) gives a 

 long and valuable illustrated account of Dr. Max Bruckner's 

 elaborate work Ueber die gleicheckig-gleichfldchiger, discontin- 

 nerlicher und nicht-konvexen Polyheder. 



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

 Greenwich. 



Stellar Clusters. — Several researches of considerable interest 

 and importance in connection with stellar clusters have recently 

 been published which are summarised below. 



A second paper on the theory of star-streaming and the 

 structure of the universe by J. H. Jeans appears in M.N., R.A.S. 

 lxxvi. 19 1 6, p. 552. In the earlier paper, summarised in these 

 notes (Science Progress, April 19 16, p. 621), he showed that 

 the observed laws of star-streaming could not be explained or 

 interpreted if it was supposed that the universe had reached a 

 steady state. He therefore seeks an explanation for them in 

 collisions between two or more clusters of stars, conceiving the 

 universe as akin originally to a gas, each molecule of which 

 is a compact cluster of stars. Such a collision will however 

 not lead to a rebound as in the case of two molecules, but the 

 clusters will intermingle and pass through one another, the 

 stars in each being affected by the attraction of those in the 

 other. A repetition of this process will gradually lead to the 

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