6i8 SCIENCE PROGRESS 



and proves two general theorems of importance about such 

 oscillation functions. 



No. 9 of the same series is an extension by Ford (ibid.) 

 of some familiar theorems concerning the relations between the 

 roots of a polynomial and those of its first derivative to the 

 more general case of the rational function with a pole at a 

 single point. 



But most of these papers concern the subject of spherical 

 harmonics, and here Whittaker's influence is very marked. 

 Edward Blades (ibid.) finds what form the function which is the 

 integrand in Whittaker's general solution of Laplace's equation 

 must have in order that the solution may be a spheroidal har- 

 monic. Whittaker (ibid.) continues his work of 191 2 and 1914 

 on integral equations corresponding to Mathieu's equation and 

 Lame's equation, of which the latter has already been noticed 

 in this Quarterly for July 191 5 (10, 120) and October 191$ (10, 

 279). In a paper on a wide class of equations of the same 

 kind (ibid.) Whittaker obtains the integral equation which is 

 satisfied by their solutions. 



The solutions of Mathieu's differential equation or the 

 equation of elliptic cylinder functions have also been considered 

 by E. Lindsay Ince (Proc. Edinb. Math. Soc. 191 5, 33). The 

 general solution of this linear differential equation is known to 

 be of a type involving two periodic functions <f> (2) and yfr (z), 

 which under certain circumstances cease to be distinct, when 

 the general solution degenerates into a single solution. Ince 

 discovers and investigates the nature of the corresponding 

 second solution. A. G. Burgess (ibid.) considers determinants 

 which give the infinite series of relations between a and q in 

 the coefficient of y in Mathieu's equation, when the solutions 

 are purely periodic. Another paper by Ince (M.N. Roy. 

 Astron. Soc. 191 5, 75, 436), concerned with the extension of 

 Mathieu's equation, known as Hill's equation in G. W. Hill's 

 Lunar Theory of 1877, gives an application of Whittaker's 

 method of solving Mathieu's equation to obtain a general 

 solution of Hill's equation. 



Archibald Milne (Proc. Edinb. Math. Soc. 191 5, 33) investi- 

 gates the disposition of the roots of the confluent hypergeometric 

 functions, and exhibits the results in a graphical form. In 

 particular the zeros of the parabolic cylinder functions are 

 discussed. 



