RECENT ADVANCES IN SCIENCE 619 



Prof. H. M. Macdonald (Proc. Lond. Math. Soc. 191 5, 14, 

 410) gives in analytical form the argument, which demonstrates 

 the theorems used in his treatment of diffraction in his Electric 

 Waves, that the constants of the requisite series are the same 

 for the solution of the diffraction problem and for the corres- 

 ponding potential problem. Dr. T. J. I'A. Bromwich {ibid. 

 450) considers the problem of the diffraction of waves by a 

 wedge by a generalisation of the process used in electrostatics 

 of taking images, by replacing the sum of the effects of n images 

 by a complex integral. 



Maurice Frechet (Trans. Amer. Math. Soc. 1915, 22, 215) 

 considers the representation of bilinear " fonctionnelles ." 



Sir Ronald Ross (Science Progress, 191 5, 10, 218 ; 191 6, 

 10, 393) gives a restatement of his researches published in 1905 

 and subsequent years on his notation for operations and the 

 solution of equations by operative division. The third part of 

 his researches appears in the present number. 



Prof. E. W. Hobson (Proc. Lond. Math. Soc. 191 5, 14, 428) 

 proves more simply Plancherel's theorem on the convergence 

 of a series of normal orthogonal functions ; and then gives a 

 proof of Weyl's result on the summability of such series in accord- 

 ance with Cesa.ro 's method, which is free from the intricacy of 

 Weyl's own method. 



Eric H. Neville (Math. Gaz. 191 5, 8, 151) gives an account 

 of his method of solution of numerical equations which has been 

 already referred to in Science Progress for October 191 5 

 (10, 279). 



Geometry. — Among papers on geometry, we will mention 

 those by Gaston Darboux on families of surfaces whose ortho- 

 gonal trajectories are plane curves (Bull. Sci. Math., 191 5, 39, 

 62), G. M. Green on the theory of curved surfaces and canonical 

 systems in projective differential geometry (Trans. Amer. Math. 

 Soc. 191 5, 22, 1), William Caspar Graustein on the equiva- 

 lence of complex points, planes, and lines with respect to real 

 motions and certain other groups of real transformations (ibid. 

 33), Virgil Snyder and F. R. Sharpe on certain quartic surfaces 

 belonging to infinite discontinuous Cremonian groups (ibid. 

 62), E. J. Wilczynski on the general theory of congruences 

 (ibid. 311), several papers by J. de Vries on the characteristic 

 numbers of systems of curves and bilinear congruences ( Versl. 

 Kon. Akad. van Wet., Amsterdam, 23, 907, 1032, 1226, 1232, 



