652 SCIENCE PROGRESS 



has at most a finite number of integer solutions. There is also another by 

 Hilbert and Hurwitz that all the rational solutions of / (x, y, z) = o, a 

 homogeneous equation representing a curve of genus zero, can be found by- 

 solving equations of the first and second degrees. 



We shall conclude by mentioning one other question of the many to be 

 found in Dickson, namely, that the sum of three biquadrates is never a 

 biquadrate, that is, there are no rational solutions of 



^* + 3^* + 2* = ^* 



apart from the obvious ones for which two of the unknowns are zero. Euler 

 stated 150 years ago that this theorem was hardly to be doubted, but he 

 could not prove it. The difficulty of the question may be gauged from the 

 fact that Prof. Dickson gives only three references to other writers. 



The above account may give the reader some idea of the contents of 

 Prof. Dickson's volume, and convince him that he should lose no time in 

 acquiring so valuable a guide to the highways and byways of the Theory of 

 Numbers. 



THE COMPLETION OF THE PUBLICATION OF THE COL- 

 LECTED PAPEHS OF THE LATE LORD RAYLEIGH, 



by Prof. Alfred W. Porter, D.Sc. : on Scientific Papers, by John 

 William Strutt, Baron Rayleigh, O.M., D.Sc, F.R.S. [Vol. VI, 

 pp. xvii + 718.] (Cambridge : at the University Press, 1920. Price 

 505. net.) 



In this sixth volume appear all the papers of the late Lord Rayleigh published 

 in the period 1911-19, together with two papers which were left ready for 

 the press, but were not sent to any channel of publication until after the 

 author's death. Those which were not printed off until after Lord Rayleigh's 

 death have been carefully revised by Mr. W. F. Sedgwick, late Scholar of 

 Trinity College, Cambridge, who has added footnotes to elucidate doubtful 

 or obscure points in the text. 



To look back upon Lord Rayleigh's work is to survey the developments 

 of mathematical and experimental physics during the last fifty years. 

 While there are subjects which exerted a special attraction to him, the chief 

 characteristic of his writings is the great range that they cover. Of the 

 grand total of 446 communications, 97 are contained in the present volume. 

 A classified index to all six volumes at the end groups the papers according 

 to subjects, and in this brief survey of the last instalment it will be convenient 

 to keep to the same classification. When a paper falls into more than one 

 group any reference to it will be made only in connection with the subject 

 to which it in the main belongs. 



This procedure practically excludes special reference to mathematics. 

 With Rayleigh, mathematics was always looking forward to its applications. 

 His general attitude is well illustrated by No. 404, " on Legendre's function 

 P„(A) when n is great and 6 has any value." After pointing out that Hobson 

 has developed the complete series proceeding by descending powers of n, 

 not only for P(m), but also for its associated functions, he adds : " The 

 generality aimed at by Hobson requires the use of advanced mathematical 

 methods. I have thought that a simpler derivation, sufficient for practical 

 purposes and more within the reach of physicists with a smaller mathematical 

 equipment, may be useful." He proceeds to supply this derivation by simple 

 methods of successive approximation. By acting in this and other ways as 

 the intermediary between the rigour of mathematics and its employment by 

 the physicist, he has often earned the gratitude of the latter, though, at 

 the same time, he may have frequently incurred the displeasure of the 

 mathematician. 



