370 Dr. Gr. J. Stoney on a Sujyposed Proof 



to by Mr. Preston, where it is demonstrated that the undula- 

 tions of plane waves are competent to form what is there called 

 the Standard Image : an image which is identical with the 

 original disturbance if the latter be one which expends all its 

 energy in the propagation of the waves. (See p. 339 of the 

 Philosophical Magazine for October 1896.) 



Accordingly the theorem may be enunciated under either 

 of two aspects. It is immaterial into which form it is put, 

 since each implies the other, so that either being established 

 both are proved. Mr. Preston adopts the second form in 

 paragraph 2 of his paper, on p. 281, where he puts the 

 enunciation into the following terms, to which I have made 

 additions within brackets which are introduced to make the 

 meaning unmistakable : — 



u Any disturbance however complex within a given region of 

 space " [^provided only that it be one which expends all its 

 energy in generating waves'] u may he resolved into a system of 

 plane-wave components'" \iohich are real]; that is, which belong 

 to the undulations actually generated in the medium that 

 pervades the space — undulations which if unobstructed spread 

 from the disturbance through the whole of space. 



Such being Theorem A, we have now to compare it, or 

 rather contrast it, with Theorem B, which Mr. Preston sup- 

 poses to be its analytical expression. Theorem B will be 

 found at p. 283 of Mr. Preston's paper, and is as follows : — 



/K y> 9 > *) = ^ A cos ip* + v.y "+ rz + st ) l / B \ 



+ 2B sin (px + gy + rz + st) / ' 



in which the coefficients, the A's and B's, have the purely 

 scalar values assigned to them at the top of the same page. 

 Accordingly vectors have no place anywhere in equation (B) ; 

 and as a consequence f(x, y, z, t) is incompetent to represent 

 the " disturbance however complex within a given region," 

 which is what we have to analyse. In fact any single ex- 

 pression, like the first or the second member of equation (B), 

 which purports to represent a "disturbance however com- 

 plex," must include the vectors of the transversals as well 

 as their scalar values. Moreover, in using equation (B) the 

 coordinates x, y, z must be restricted to points within the 

 " given region of space." The supposition then that the scalar 

 equation given by Mr. Preston can possibly be the analytical 

 expression of Theorem A, falls to the ground. 



It may at first sight appear as if these difficulties could be 

 met by the familiar expedient of representing the motion 

 within a given space not by the one function J?(x,y, z, t) 

 which is not scalar, but by three purely scalar equations, 



