266 J. W. Gibbs—Double Refraction and the 
7. The kinetic energy of the whole medium is represented 
by the double volume-integral* 
gaff fEFO ere), dee. 
where dv,, dv, are two infinitesimal elements of volume, (E+é’, 
(€+€’), the corresponding components of flux, 7 the distance 
between the elenfents, and ¥ denotes a summation with respect 
to the codrdinate axes. Separating the integrations, we may 
write for the same quantity 
3/(E+2), | f See do, | dv.. 
It is evident that the integral within the brackets is derived 
from €+€ by the same process by which the potential of any 
mass is derived from its density. If we use the symbol Pot to 
express this relation, we may write for the kinetic energy 
4E/E+E) Pot (E+E') dv. 
The operation denoted by this symbol is evidently distribu- 
tive, so that Pot (E+€’)= Pot &+Pot é’. The expression for the 
kinetic energy may therefore be expanded into 
43 /é Pot & dv+42/é Pot & dv 
4435/8! Pot F dv+4>/& Pot &' de. 
But & , and therefore Pot é/ , has in every wave-plane the 
average value zero. Also and therefore Pot &, has in every 
wave-plane a constant value. Therefore the second and third 
integrals in the above expression will vanish, leaving for the 
kinetic energy 
42> /é Pot & dv+42>/& Pot &' dv, (3) 
which is to be calculated for a time of no displacement, when 
2a 3 tae ‘5 any U4 
ates al cos an; ote cos an, 6 cos 2n-. ( ) 
The form of the expression (3) indicates that the kinetic 
energy consists of two parts, one of which is determined by the 
regular part of the flux, and the other by the irregular part of 
the flux. 
* The fl tic system of 
units. It is to be observed that the difference of opinion which has prevailed 
circuits. 
