208 Dr. G, J. Stoney on Proofs of 



uniform plane waves, and is the principal one of a definite 

 group among them the members of which are characterized 

 by having no bearing upon the physical theorem. 



Again, on p. 283, Mr. Preston gives a scalar equation 

 which he speaks of as " the analytical expression of the 

 general theorem enunciated by Dr. Stoney." On p. 371, I 

 pointed out that the first step towards correcting this oversight 

 is to substitute three expressions for £, v, and % for Mr. Pres- 

 ton's single expression. This Mr. Preston does in his subse- 

 quent letter in the June number of the Magazine, p. 458. 

 The further steps necessary to complete the correction are : 

 (1) To add the proof referred to above in the third paragraph 

 of this letter ; and (2) to embrace the entire of space in the 

 investigation. 



This naturally leads to the remark that when Mr. Preston's 

 investigation is confined to a limited area round the originating 

 disturbance it furnishes a different resolution. This is a 

 kinematical * resolution ; and the kinematical resolution of 

 this Fourier analysis is only one of innumerable resolutions 

 of a like kind that are possible. Attention was called to their 

 unlimited number in my first paper on Microscopic Vision. 



It is very instructive to trace out what the physicist is to 

 understand by that particular resolution which the Fourier 

 analysis offers to us, when it is made applicable to space of 

 three dimensions and when it is restricted to a defined area. 

 This can be done. Let the whole of space be called A ; 

 let the "prescribed area" (which includes the originating 

 disturbance and a limited region about it) be called B ; and 

 let the rest of space be called C. Then B + C = A. Let now 

 the radiations from the originating disturbance (which may 

 spread to any distance) be resolved into trains of uniform 

 plane waves of infinite extent laterally and travelling with 

 the velocity proper to the medium. These waves can 

 advance unchanged through the medium, and are a body of 

 uniform plane waves which we shall call (a). This resolution 

 is unique. It is that special one which results from the 

 medium being agitated by the originating disturbance only. 

 It may be called the legitimate resolution of radiations from 

 that disturbance. This complete system of undulations may 

 be distinguished into two parts : (a^) the part of it which falls 

 within the region B, and (a 2 ) the part which lies outside. 



* By a kinematical resolution is to be understood a resolution into 

 undulations (in this case of limited extent) which the medium is incom- 

 petent to transmit forward without change, and, moreover, of which no 

 one could be propagated even within the prescribed area, if isolated from 

 the others. 



