RECENT ADVANCES IN SCIENCE 353 



at interior points, as well as at the end points, of the interval 

 over which the equation is considered (cf. Wilder's paper men- 

 tioned in Science Progress, 191 8, 12, 547). 



W. D. MacMillan (Trans. Amer. Math. Soc. 191 8, 19, 205-22) 

 investigates the reduction of certain differential equations of 

 the second order. 



H. J. Ettlinger (ibid. 79-96) extends the methods of Bocher 

 (1905) and Birkhoff (1909) — which are based on the application 

 of Sturm's theorems — to the most general, real, self-adjoint 

 linear system of the second order. 



W. E. Milne (ibid. 143-56) discusses the degree of converg- 

 ence of the series (treated by Birkhoff in 1908) built up out of 

 solutions of an ordinary linear differential equation. 



T. H. Hildebrandt (ibid. 97-108), in connection with his 

 paper referred to in Science Progress (1917, 12, 12), discusses 

 boundary-value problems in linear differential equations in 

 general analysis. 



M. J. M. Hill (Proc. Lond. Math. Soc. 191 8, 17, 149-83) 

 investigates the singular 1 solutions of ordinary differential 

 equations of the first order with transcendental coefficients. 



Hill's classification of the integrals of linear differential 

 equations of the first order (Science Progress, 191 8, 12, 548) 

 should be compared with that independently introduced by 

 H. Bateman's book, to be reviewed in a future number. 



Bateman (Bull. Amer. Math. Soc. 191 8, 24, 296-301) dis- 

 cusses the solution of the wave-equation by means of definite 

 integrals. 



Questions in the solution of integral equations are treated 

 by A. Korn (Archiv der Math. 1916, 25, 148-73) and A. Hoborski 

 (ibid. 200-2). Major P. A. MacMahon and H. B. C. Darling 

 (Proc. Camb. Phil. Soc. 191 8, 19, 178-84) obtain some interesting 

 reciprocal relations in the theory of integral equations. 



W. H. Wilson (Amer. Journ. Math. 191 8, 40, 263-82) under- 

 takes a systematic theory of a very general addition formula 

 for functional equations. 



Geometry. — M. Pasch (Journ. fiir Math. 147, 184-90) dis- 

 cusses fundamental questions in geometry ; L. Berwald (Sb. 

 Munchen, 191 6, 1-18) treats curves that are algebraically 

 rectifiable in non-Euclidean space ; and E. Study (Leipzig Ber. 

 1 91 6, 68, 65-92) considers the " principle of the conservation 

 of number " in enumerative geometry. 



