of a Theorem hi Wave-motion. 373 



terms of which represent mere artificial undulations which 

 advance in only two directions, one perpendicularly forwards 

 the other perpendicularly backwards across the disk, which 

 may not be carried one step outside it, which inside it are of a 

 kind that no medium could propagate, which are in fact a mere 

 mathematical fiction, and not any physical analysis whatever 

 of events going on in nature. Yet these two analyses, one 

 by Theorem A the other by Fourier's Theorem, though so 

 utterly unlike, are identified with one another in the first of 

 the two statements made in the third paragraph of Mr. Preston's 

 paper. 



In the paper preceding Mr. Preston's it is shown on p. 273 

 that Theorem A may be expressed symbolically by the equation 



F (x, y, z, = jj 2 f"M sin Utt ^? + «)] . sin d6 dfi (A) 



where r~ x cos +y sin cos $ + z sin sin <f>, and in which 

 the M's are directed quantities. The values of v and the 

 vector components of the M's depend on the properties of the 

 medium, and may be expressed as functions of and <f> when 

 we know the equation of the wave-surface in the medium. 

 On the other hand the a's and the scalar components of the 

 M's depend on the originating disturbance. Now what is 

 wanted is such an analytical proof of Theorem A as will give 

 us symbolical expressions for these quantities as functions of 

 and cf> ; and it may be hoped that Mr. Preston, with his 

 experience in dealing with this class of problem, will yet be 

 able to substitute the really valuable proof which will accom- 

 plish this for the illusory proof which, on a first view, he has 

 mistaken for it. 



I have made a slight attempt by adopting polar coordinates, 

 but hitherto without success, to find some use for Mr. Preston's 

 extension of Fourier's Theorem. It is perhaps not impos- 

 sible that a fuller search in this direction may bear fruit. 

 But whatever the issue, it is plain that Mr. Preston's extension 

 of Fourier's Theorem, though it may be of limited applicability 

 in physics, is of interest as a mathematical theorem. 

 I am, Gentlemen, 



Faithfully yours, 



G. Johnstone Stoney. 

 8 Upper Hornsey Rise, N., 

 April 12, 1897. 



