370 SCIENCE PROGRESS 



of a certain kind of linear homogeneous ordinary differential 

 equation is in general asymptotic, in the sense of Poincare, to 

 a diverging power series in a sector of the complex plane in 

 which it is analytic, the asymptotic representation being valid 

 for the variable approaching infinity in an appropriate direction. 

 A step towards the ideal of obtaining a single expansion of such 

 character that it is capable of exhibiting the asymptotic pro- 

 perties of a function near infinity and of yielding at the same 

 time a convenient and workable representation of it in the 

 finite part of the plane is given by factorial series and certain 

 immediate generalisations of them (Norlund, 1914). In 1916, 

 R. D. Carmichael [Trans. Amer. Math. Soc. 191 6, 17, 207-32 ; 

 Science Progress, 191 7, 11, 456-7) pointed out that such 

 factorial series are instances of a large class of series with simple 

 properties, and in the Bull. Amer. Math. Soc. for 191 7 (23, 

 407-25) he showed how several important series in mathe- 

 matical literature are included as special cases of series of the 

 form just referred to, and in which the nth. term is a constant 

 multiplied by g (x + n)/g(x), where g(x) is a given function of 

 x. A further generalisation of this series is also considered. 



H. S. Carslaw (Proc. Lond. Math. Soc. 191 7, 16, 84-93) 

 shows, in a second paper on the Green's function for Poisson's 

 equation, that, with the same assumptions as are involved in the 

 usual treatment of the differential equations of mathematical 

 physics, the Green's functions in question can be obtained im- 

 mediately from an integral equation, when the region with which 

 we are dealing is finite. 



G. B. Jeffery (ibid. 133-9) shows that Whittaker's general 

 solution of Laplace's equation provides a ready means of 

 expressing a given potential function in terms of different 

 harmonics — spherical, cylindrical, and spheroidal. 



Geometry. — J. E. Rowe (Bull. Amer. Math. Soc. 191 7, 23, 

 405-7) investigates the projection of a line section upon a 

 rational plane cubic curve. 



A. B. Coble (Trans. Amer. Math. Soc. 191 7, 18, 331-72) 

 gives the third part of his investigations on point sets and 

 allied Cremona groups. The first two parts were published in 

 the same Transactions for 191 5 and 1916. 



F. R. Sharpe and V. Snyder (ibid. 402-14) obtain a classi- 

 fication of the possible (2, 2) point correspondences between two 

 planes, and describe the important features of each type. 



