264 SJ. W. Gibbs—Double Refraction and the 
the regular part of the displacement may be represented by the 
equations 
4 t 
§=a cos ann cos on ) 
t 
n= f cos one cos wear (1) 
u t | 
= § 27— 27— 
6-9, 00 7 ©8 i? J 
which passes through the origin. Since u is a linear function 
of «, y, and z, we may regard these equations as giving the 
values of &, 7, C, for a given system of waves, in terms of a, y, 
z, and ¢. 
and secondly one-half the difference, of the original motion an 
that obtained by substitution of —é for ¢, we may separate the 
non-synchronous part of the irregular oscillations from the rest 
of the motion. Therefore, the supposed non-synchronous part 
of the irregular displacement, if capable of existence, is at least 
wholly independent of the wave-motion and need not be con- 
sidered by us. 
