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LIX. On a Supposed Proof of a Theorem in Wave-Motion. 

 To the Editors of the Philosophical Magazine. 

 Gentlemen, 



IN a letter printed at page 368 of this volume, under the 

 above title, Dr. Stoney directs attention to a communi- 

 cation* in which I expressed the possibility of expanding any 

 function of any number of variables in the form 



/(#, y . . .) = 2A cos {lx + my + . . .) 



+ 2Bsin {lr + my + ...). . (1) 



He admits this to hold good for " scalar functions," but for 

 such functions only ; and he concludes that the applications 

 of it " to prove certain physical theorems which treat of real 

 wave-motions " are erroneous. 



In reply, I may state first of all that I fear Dr. Stoney 

 has misunderstood my communication. What I intended 

 to convey was, that if we take the variables to be a?, y, z, t — 

 namely, the coordinates of a point in space and the time — 

 then the above expansion enables us to resolve any specified 

 disturbance, existing throughout any given region, into a 

 system of simple harmonic plane waves. When I say a 

 specified disturbance, I mean simply that the disturbance at 

 every point of the region is expressed in the ordinary way in 

 terms of its three component velocities or displacements in 

 the form 



f =-I , 1 (* l y,«,*)j V = ¥ 2 {ie,y,z,t), ?= F z (x,y,z,t), 



and I did not think it necessary to state such an obvious 

 proceeding. Each of these functions can be expanded, as 

 Dr. Stoney admits, and the components of the simple har- 

 monic plane waves are then to hand. 



Of course a velocity (or a displacement) is a directed 

 quantity, and a function which represents a velocity is a 

 vector function ; but it is here a vector function of scalar 

 variables, and accordingly the analysis which I have employed 

 holds good. In fact, if F be such a vector function, we may 

 write it in the form 



F = iF.+JF.+kF,, 



where Fj, F 2 , and F 3 are scalar functions which, by admission, 

 may each be expanded in the form (1), so that F is thrown at 

 once into a sum of simple harmonic terms of the type 

 (iAj +j A 2 + h A 3 ) cos [lx -f my + . . .) = A cos (la + my + ...), 



* li On the General Extension of Fourier's Theorem," p. 281 of this 

 volume. 



