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XLVIII. On a Supposed Proof of a Theorem in Wave-motion. 



To the Editors of the Philosophical Magazine. 



Gentlemen, 

 T p. 281 of this volume of the Philosophical Maga- 



A 



zine, Mr. Thomas Preston announces an extension of 

 Fourier's theorem whereby a scalar function of any number 

 of variables may be expanded in a series of sines and cosines of 

 linear functions of the variables, with coefficients expressible 

 as definite integrals. This remarkable theorem is of much 

 interest from the mathematician's point of view, and is likely, 

 when handled correctly, to be of use to physicists also. But, 

 in applying it, it is essential that we should attend to the fact 

 that it deals with scalar quantities only, and cannot be 

 employed in physical problems except so far as the mathe- 

 matical discussion of the problem, throughout whatever is its 

 extent in space, admits of being brought into a scalar form. 

 Another consideration must also be kept in mind, viz. : that 

 what presents itself in optics as diffraction is encountered 

 in every real wave-motion. Both of these considerations, 

 and the distinction between forced wave-motion and wave- 

 motion which the medium is able to propagate, seem to have 

 been overlooked in two attempts which Mr. Preston has made 

 to employ his theorem to prove certain physical theorems 

 which treat of real wave-motions, one attempt on p. 283 and 

 the other on p. 285 ; and unfortunately the oversights 

 vitiate the proofs which he offers. They also affect what is 

 said in the third paragraph of his paper, on p. 281. This 

 paragraph contains two statements, of which the first is erro- 

 neous owing to the omission of these considerations, and the 

 second can only be rendered correct by interpreting the word 

 " function " to mean a function containing vectors as well as 

 scalars, in which case the observation although true would 

 have no relation to what follows it. 



In the first paragraph of his paper, on p. 281, Mr. Preston 

 quotes the enunciation of a theorem in wave-propagation as 

 fol!ov>s: — 



"However complex the contents of the objective field, and 

 whether it or parts of it be self-luminous, or illuminated in 

 any way however special, the light which emanates from it 

 may be resolved into undulations each of which consists of 

 uniform plane waves." 



This enunciation is taken from a paper on Microscopic 

 Vision, and is in reality a general theorem in wave-motion 



