RECENT ADVANCES IN SCIENCE 269 



T. W. Chaundy and A. E. Jolliffe (Proc. Lond. Math. Soc. 

 1 91 6, 15, 214) obtain the remarkably simple necessary and 

 sufficient condition for the uniform convergence throughout any 

 interval whatever of the series Xa n sin nO, where (a n ) is a sequence 

 of positive numbers decreasing steadily to zero, is that na n 

 tends to zero. 



M. B. Porter (Proc. Nat. Acad. Set., Washington, D.C., 

 1916, 2, No. 4) gives a simple proof of Lucas's theorem that the 

 zeros of any polynomial F'(z) lie inside any closed convex 

 contour inside which the zeros of F(z) are, and the theorem is 

 extended to give information concerning the distribution of 

 zeros of the derivative of certain rational or transcendental 

 functions (cf. also ibid. No. 6). 



Lord Rayleigh, in a paper read to the Royal Society on 

 May 11, 1 916, gave an approximate formula for Legendre's 

 function P n (0), when n is great and 6 has any value, which is 

 sufficient for practical purposes and whose derivation is more 

 within the reach of ordinary physicists than is Hobson's 

 elaborate mathematical investigation. 



W. L. Miser (Trans. Amer. Math. Soc. 1916, 17, 109) con- 

 tinues the work of Picard (1880) and others on linear differential 

 equations having elliptic function coefficients ; Miser, however, 

 investigates multiform solutions. 



G. M. Green (Proc. Nat. Acad.Sci., Washington, D.C., 1916, 

 2, No. 4) generalises the theory of linear dependence to the case 

 of n functions of several independent variables and applies it 

 to the study of certain completely integrable systems of partial 

 differential equations. 



W. L. Hart (ibid. No. 6) treats three problems : (1) Certain 

 fundamental theorems concerning a type of real-valued func- 

 tions of infinitely many real variables ; (2) The problem of 

 infinite systems of ordinary differential equations ; (3) The 

 fundamental problem of the theory of implicit functions in this 

 field. 



Continuing some work of Wilczynski, E. B. Stouffer (Proc. 

 Lond. Math. Soc. 191 6, 15, 217) is concerned with the problem 

 of calculating the seminvariants for a system of linear homo- 

 genous differential equations. 



Dr. H. Bateman, in a paper read to the Royal Society of 

 Edinburgh on July 3, 191 6, showed how the general equation 

 of wave-motion associated with Maxwell's electromagnetic 

 18 



