MATTER, A MODE OF MOTION 357 



would be three regions. Near the center would be a relatively small core in 

 which the non-linear effects predominate and linear theory is totally inap- 

 plicable. Farther out the departure from linearity is only moderate, and the 

 variation of the constants with distance is slow enough that the reflections 

 are small. It should be possible to treat wave propagation in this region by 

 the methods developed for a string of variable density, which are sometimes 

 cited as analogous with those employed in wave mechanics. The analogy is 

 made closer by the fact that the variations in impedance which correspond 

 to the varying density are determined by the energy density of the pattern 

 itself. Still farther out the amplitudes become still smaller, the ether con- 

 stants become very nearly but not quite uniform, and the pattern ap- 

 proaches very closely to that in a linear medium. 



While the nature of the pattern is determined largely by the non-linear 

 inner region, because of the small volume of this region most of the energy 

 will be located in the nearly linear region. So we might expect some at least 

 of the macroscopic properties of the pattern to differ very little from those 

 deduced from a consideration of the corresponding pattern in a linear me- 

 (lium. We will therefore begin by examining such a pattern. For the linear 

 case, when the axes are at rest with respect to the undisturbed ether, (1) 

 and (2) lead to the wave equation for the vector displacement s, 



^'' c'v'-s. (3) 



5 s 2„2 



As is well known, this is satisfied by any function of the form 

 where 



— Rx \ "y I Rz J \^) 



and the constants co, ^x, ^y and h^ , are real or complex. Since an imaginary 

 frequency is interpreted as an exponential change with time, it is not suit- 

 able for representing a permanent pattern, so co will be taken to be real. 

 Imaginary values of k are interpreted as exponential variations with dis- 

 tance. But, since s is always real, we may, by a four-dimensional Fourier 

 analysis, represent/ as the summation of components of the form 



s = J^ cos (co/ ± kxX ± kyy ± k^z), (5) 



where A is a, complex vector representing the amplitude and phase of the 

 component, and kx , ky and k^ are real. Since each component must satisfy 

 (3), the new constants must satisfy (4). Each such component constitutes 

 a plane progressive wave traveling, with velocity c in a direction, the cosines 

 of which are proportional to the wave numbers kx , etc. 



