MATTER, A MODE OF MOTION 359 



In order to derive a quasi stationary pattern we replace the functions 

 g+( ) and g_( ) by Bcosa{ ) and combine components in a manner 

 similar to that used in deriving (6). The result is 



S = 9,B cos a/^co ( ' ~ — -^^ ) cos a:/3^i(.v — Vt) cos akyj cos ak^z, (7) 



where 5 is a complex vector, and a may be any real scalar function of V. 

 When we compare this with (6) we tind that the last three factors, which 

 in (6) describe a fixed envelope, in (7) describe an envelope which moves in 

 the .V direction with velocity I'. For the same values of k^ , k^j and k^ , the 

 moving pattern has its dimensions in the x direction reduced relative to 



those in the v and z in the ratio -. The first factor in (6) describes a sinusoidal 



variation with time which is everywhere in the same phase. In (7) it de- 

 scribes one, the phase of which varies linearly with .v. This factor also de- 



scribes a wave which progresses in the .v direction with a velocity — . The 



existence of such a wave as a factor in the expression for a moving wave 

 pattern was commented on by Larmor."* Aside from the constant a in (7) 

 it will be recognized as the Lorentz transform of (6), as it should be since 

 the approximate equations of which it is a solution are invariant under 

 this transformation. 



We shall take (7) to represent one component of a moving wave pattern 

 which represents a moving particle. If we transform this to axes moving 

 with the pattern by a Newtonian transformation it becomes 



s = SB cos oc { - t' — — r- x' I cos q;/3^j x' cos aky y' cos ak^z', (8) 



in which the envelope is at rest. This may be thought of as a stationary wave 

 in an ether which is movmg relative to the axes with a velocity — V. It 

 is a solution of the wave equation for such an ether, as obtained by trans- 

 forming (3) to the moving a.xes, or 



dt'- dx'dt' dx- 



The one dimensional form of this equation is identical with that given by 

 Trimmer for compressional waves in moving air, except that in one case s 

 is solenoidal and in the other divergent. 



So far we have found no reason to associate any particular moving 

 pattern with the assumed stationary one, in the sense that the moving pat- 



^Larmor, Ency. Brit. 11th Ed., 1910; 13th Ed., 1926, Vol. 22, p. 787. 

 ^ J. D. Trimmer, Jour. Aeons. Sac. Am., 9, p. 162, 1937. 



