210 On Proofs of a Theorem in Wave-motion. 



disturbance operating alone upon an actual physical medium. 

 It is therefore the one which should be employed by the 

 physicist. 



Mr. Preston's method of analysis is very much to be 

 welcomed, both as extending Fourier's analysis with all its 

 kinematical merits to two, three, and more dimensions, and 

 as one which, ivhen correctly handled, can furnish an alternative 

 proof of an important physical theorem by which it has been 

 possible to unravel the very intricate phenomena of micro- 

 scopic vision "*. It is perhaps disappointing to find that the 

 analytical proof has not as yet added anything to the knowledge 

 of nature which had previously been obtained by the geome- 

 trical proof. This from the mathematical standpoint appears 

 to be because the definite integrals of the analytical proof rest 

 upon data in which the character of the originating disturbance 

 and the properties of the medium are not kept separate. If a 

 proof is found in which they can be kept distinct, either by 

 combining a physical theorem with Mr. Preston's extension 

 of Fourier's kinematical theorem, or otherwise ; this is the 

 kind of analytical proof which may be expected to add some- 

 thing to our knowledge of nature. 



The whole inquiry is a striking example of the great 

 advantage of pushing geometrical and analytical proofs for- 

 ward side by side. We thus gain, as was distinctly seen 

 during MacCullagh's lifetime (see July Magazine, p. 98), a 

 deeper and more accurate knowledge of what is going on 

 in nature than is within our reach if we restrict ourselves to 

 the analytical method, as is too much the practice. In the 

 present instance the geometrical proof is the more direct and 

 places clearly before us the physical connexion of the actual 

 events with one another. It was therefore not unnaturally 

 the method of proof by which the theorem was discovered. 

 On the other hand the Fourier method, while it leaves these 

 physical relations out of sight, brings new kinematical relations 

 into prominence and forces upon our attention how largely 

 the further progress of events in nature when once set going 

 admits of such purely kinematical expression as links the 

 actual events of nature with events which though physically 



* Typical specimens of all the principal phenomena, except those that 

 concern black-field illumination, are dealt with in three papers on 

 " Microscopic Vision," in the Phil. Mag. for October, November, and 

 December, 1896. When black-field illumination is employed the image 

 is produced in one or other of two distinct ways, of which a study has 

 since been made by the present writer. He hopes soon to have time to 

 publish it, and this will apparently complete the explanation of micro- 

 scopic vision. 



