RECENT ADVANCES IN SCIENCE 549 



R. A. Johnson (ibid. 313-17) applies the method of directed 

 angles which he used earlier in 191 7 to the theory of inversion, 

 especially to the use of this theory in studying properties of the 

 triangle. None of the theorems which he obtains is new, but 

 in most cases the form of presentation is new and has certain 

 other advantages. 



O. J. Peterson (ibid. 376-9) proves, without the use of 

 Pliicker's formulae, the fundamental theorem of Clebsch (1864) 

 that every rational curve of the nth order has \(n— i)(n— 2) 

 double points. In all his paper, Peterson assumes that all 

 singular points which occur are either nodes, conjugate points, 

 or cusps. He then shows that the upper limit of the number 

 of cusps is §(«— 2). 



G. M. Green (Trans. Amer. Math. Soc. 1917,18, 480-88) gives 

 two geometric characterisations of isothermal nets on a closed 

 surface ; and P. F. Smith (ibid. 522-40) gives a theorem for 

 space analogous to Cesaro's theorem in his Geometria intrinseca 

 of 1896 for plane isogonal systems. 



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

 Greenwich. 



The Galactic Distribution of the Stars. — Until recently, our 

 knowledge of the galactic condensation of stars of different 

 magnitudes was one of extreme uncertainty. Whilst all in- 

 vestigators agreed that stars of all magnitudes were condensed 

 towards the galaxy, widely differing values were obtained by 

 various investigators of the amount of this condensation and 

 of its rate of variation with magnitude. Three papers recently 

 published may be said to have removed this uncertainty, and 

 to have settled beyond dispute the order of magnitude of these 

 fundamental data of stellar distribution ; these papers are by 

 van Rhijn (Groningen Publications, No. 27, 191 7), Seares 

 (Astroph. Journ. 46, 117, 191 7), and Chapman (M.N., R.A.S. 

 78, 66, 191 7). The two principal previous determinations 

 were by Kapteyn (Groningen Pubs. No. 18, 1908) and by 

 Chapman and Melotte (Mems. R.A.S. 60, Pt. 4, 1914), the 

 latter being based mainly upon counts of photographic magni- 

 tudes on the Franklin-Adams chart plates, a series of plates 

 on a uniform scale covering the whole sky. A number of 

 regions on each plate were counted. Kapteyn 's results were 

 based on counts of visual magnitudes. Measuring the galactic 



