On a Supposed Proof of a Theorem in Wave-motion. 369 



though presented there in its optical form only. It may be 

 freed from limitation while keeping the enunciation sub- 

 stantially the same, and then becomes : — 



If space be occupied by any uniform medium capable of 

 propagating waves, and if a disturbance however complex is 

 called into existence within a defined portion of the space, then 

 the undulations ivhich emanate from the disturbance and spread 

 through the rest of space may be resolved into undulations of the 

 medium each of ivhich consists of uniform plane waves. 



This may be called Theorem A. It is a theorem which the 

 present writer proved geometrically and of which a symbolical 

 proof is much wanted. A correct symbolical proof, when- 

 ever it is discovered, will probably lead to useful determi- 

 nations of the coefficients of the expansion in Theorem A, 

 involving vectors as well as scalars, but otherwise somewhat 

 like (though necessarily more complex than) the expressions 

 for the coefficients of a Fourier's expansion. To accomplish 

 this would be an important service. Mr. Preston endeavours 

 to supply this desideratum in the paper now under discussion. 

 Each of the plane waves of Theorem A has been shown by 

 the present writer to be of unlimited extent in its plane and 

 uniform over its whole extent. It is therefore a wave which as it 

 travels forward remains unchanged whatever be the medium 

 that occupies the space, provided only that the medium be 

 uniform. Observe also that what the waves are does not 

 depend exclusively upon what the disturbance is, but upon 

 this in conjunction with the physical properties of the medium. 

 Accordingly any expressions for the coefficients of the terms 

 of the expansion which supposes these coefficients to be 

 functions of the disturbance only, such as the values given by 

 Mr. Preston at the top of p. 283, must be wrong. They can 

 only belong to some kinematical resolution consisting of 

 forced vibrations ; a mathematical exercise of little use in 

 physics, since it supplies no information about the real 

 resolution into plane waves effected by nature which is what 

 is dealt with in Theorem A. 



Of the consequences of Theorem A, that which is of most 

 value to the physicist, is that the radiations outside the region 

 of disturbance resolve themselves into undulations of uniform 

 plane waves. It is of somewhat less importance to the 

 physicist, though equally true, that the theorem also resolves 

 the disturbance itself into these same undulations, if the dis- 

 turbance be of such a kind that it expends all its energy in 

 propagating waves. (See the condition numbered 3, on page 

 141 of this volume.) The truth of this second part of the 

 theorem is shown in the paper on Microscopic Vision referred 



