a Theorem in Wave-motion. 209 



Then (a) = (a 1 + a 2 ). Hence (aj is a resolution of the actual 

 motion going on within the region B. Now we find that the 

 Fourier analysis when restricted to the region B, furnishes 

 another resolution, of which, all the waves differ from the 

 corresponding waves of (e^). This Fourier system of plane 

 waves we may call (6). Suppose each wave of the Fourier 

 system to be extended without limit laterally, and let the 

 aggregate of these extensions be called (c) . They occupy the 

 space 0. Then, although (b) consists of waves which a physical 

 medium cannot propagate, {b + c) is a system of waves which 

 the medium can propagate. The question now is, What 

 motion in the medium, or in space, does (b + c) represent? 

 Let us call it X. We know that if we confine our survey to 

 the region B, {b + c) will represent within that space the same 

 motion as (a) represents. Hence X would result from any- 

 thing which maintains the waves (aj within the space B 

 and the waves (c) outside it : in other words, it is the same 

 motion as w r ould result from maintaining the originating 

 disturbance with its actual radiations throughout an infinite 

 medium, and by adding on the kinematical motion (c — a 2 ) in 

 the space C. It is this artificial complex* which the Fourier 

 analysis represents. 



It is further noteworthy that any other arbitrary motions 

 might have been set up within the space C, and that every 

 change made in these w T ill alter the systems of plane waves 

 which sweep across the space B, so that the particular set of 

 undulations furnished by the Fourier analysis is not unique. 

 There are numberless others t- 



Another way of viewing the subject is that any system of 

 uniform plane waves of infinite extent laterally, of which the 

 resultants are cypher resultants at all points within the space 

 B whatever they may be elsewhere, may, from a mere kine- 

 matical point of view, be superposed upon the legitimate 

 resolution ; thus producing an unlimited number of kine- 

 matically possible resolutions so long as we do not look 

 beyond the region B. But among these (a) is the only 

 resolution which is physically legitimate, in the sense of 

 being the only one which is due to the action of the originating 



* This complex may be presented under an exclusively physical 

 aspect by taking into account the considerations referred to in the foot- 

 note on p. 275 of the April Magazine. But to do this would have 

 required so much explanation that, in the text, for brevity, the subject is 

 dealt with in a way which does not carry the analysis so far and 

 leaves some of the motion kinematical. 



t These others are not beyond the reach of the Fourier analysis, if 

 artificially manipulated. 



FHLMag. S. 5. Vol. 44. No. 267. August 1897. Q 



