RECENT ADVANCES IN SCIENCE 279 



the application of the method to certain equations of astro- 

 nomical interest. 



It is well known that Abel gave a certain integral equation 

 as a generalisation of the problem of the isochrone, stating 

 that he had solved it, but the solution does not appear in his 

 published works. The Rev. P. Browne (Proc. Roy. Ir. Acad. 

 191 5) finds a solution of Abel's equation which involves integra- 

 tion along an infinite straight line in the plane of the complex 

 variable ; and extends the method to the solution of certain 

 other functional equations. 



In 1902, Prof. E. T. Whittaker discovered a simple integral 

 representation of the general solution of Laplace's equation — 

 a form in which most of the well-known particular solutions 

 can be readily expressed. But the ellipsoidal harmonics, 

 which are products of Lame's functions, resisted all attempts 

 to express them in this form. However, Whittaker (Proc. 

 Lond. Math. Soc. 191 5, 14, 260) has now succeeded in doing 

 this by his discovery of a certain integral equation of which 

 the solutions are solutions of Lamp's differential equation. 



Eric H. Neville gives (ibid. 308) a new method of solving 

 simultaneous numerical functional equations, and illustrates 

 it by the solution of the amusing puzzle as to how a circle may 

 be completely covered by five smaller equal circular discs. 



We will now pass to the theory of series and the closely 

 connected theory of analytic functions. J. Kampe de FeYiet 

 (Compt. Rend. 191 5, May 3) has given a generalisation of the 

 series of Lagrange and Laplace. Prof. J. Pierpont's Functions 

 of a Complex Variable is reviewed elsewhere in this number. 



G. H. Hardy (Proc. Lond. Math. Soc. 191 5, 14, 269), acting 

 on a suggestion made by Dr. H. Bohr and Prof. E. Landau, 

 proves that the mean value of the modulus of an analytic 

 function is, like the maximum of this modulus on a circle of 

 radius r, a steadily increasing function of r. Hardy proves 

 this and much more, and the importance of such researches is 

 well known through some of the work of Poincare, Hadamard, 

 Borel, Blumenthal, and many others. 



Geometry . — The history in J apanese mathematics of the Pytha- 

 gorean equation connecting the sides of a right-angled triangle 

 is discussed by Kiochiji Yanagihara (Tohoku Math. Journ. 6). 

 Various methods of obtaining integral solutions of this equation 

 were given in the eighteenth and nineteenth centuries, and the 



