a Theorem in Wave-motion. 207 



It is to be observed that this may also be proved analyti- 

 cally — a circumstance not referred to by Mr. Preston. Some 

 proof on this head is essential, because Mr. Preston's extension 

 of Fourier's theorem has the unlimited generality of the 

 original theorem ; it applies indifferently to every motion in 

 space that can in any way be represented mathematically, 

 and takes no notice of whether it is physically or even geo- 

 metrically possible. When a proof is given that the three 

 impossible waves of Mr. Preston's equations give a resultant 

 which is a possible wave, when his treatment of the physical 

 problem receives other modifications, and when the whole of 

 space is included under it, it can be made to furnish an 

 alternative proof of what had previously been ascertained by 

 MacCullagh's more direct method, viz.: that the radiation 

 from a disturbed portion of a uniform medium which is capable 

 of propagating waves, can be resolved into undulations of uniform 

 plane waves which the medium can transmit forward unchanged. 



This, however, is not what Mr. Preston put forward in the 

 paper in which he announced his important extension of 

 Fourier's theorem. He also introduced two statements to 

 which I reluctantly found myself obliged to demur, as in both 

 he erred with regard to my work. I had, in October 1896, 

 proved a theorem (quoted by Mr. Preston on p. 281 of the 

 April Magazine) concerning the radiations from a region of 

 disturbance. About this, Mr. Preston on that page states that 

 Fourier's theorem applied " between assigned limits" to "a 

 disturbance which is a function of a single variable " is in 

 that case the analytical expression of my theorem [which 

 concerns the radiations from the disturbance] . This mistake I 

 pointed out in the following number of the Philosophical 

 Magazine, and gave a case on p. 321 in which it is manifestly 

 not true. On this Mr. Preston observes on p. 459 of the 

 Jane Magazine that after resolving the originating distur- 

 bance into one of the numberless resolutions into uniform 

 plane waves which are possible, " nothing more remains to be 

 done. As to what happens outside the disk, this is quite 

 another question." Precisely so: and as it is with this "quite 

 other question " that my theorem is concerned, associated 

 too with another of the possible resolutions within the disk, it 

 became necessary to point out that Mr. Preston had here fallen 

 into the fallacy known as ignoratio elencht, where the proof of 

 one thing is mistaken for the proof of another. Moreover, it 

 is important to emphasize, since a mistake might easily be made 

 on this head, that the resolution loithin the disk into a simple 

 Fourier's expansion is only one of innumerable ways in which 

 the motion within the disk may be resolved into trains of 



