268 J. W. Gibbs— Double Refraction and the 
al’ 2 2 2 
pom eho (8) 
9. With respect to the kinetic energy of the irregular part 
of the flux, it is to be observed that, since é, 7’; ¢’ have their 
average values zero in spaces which are very small in compari- 
son with a wave-length, the integrations implied in the notations 
Pot & Pot 7, Pot e may be confined toa sphere of a radius 
which is small in comparison with a wave-length. Since within 
such a sphere & 7, ‘a are sensibly determined by the values 
of é, om c at the center of the sphere, which is the point for 
which the value of the potentials are sought, Pot &’, Pot 7', 
Pot a must be functions—evidently linear functions—of é, ’; C; 
and €’ Pot &’, 7’ Pot 7’, ¢’ Pot ¢’ must be quadratic functions 
of the same quantities. But these functions will vary with the 
position of the point considered with reference to the adjacent 
molecules. 
Now the expression for the kinetic energy of the irregular 
part of the flux, 
$/&' Pot & dv, 
indicates that we may regard the infinitesimal element dv as 
having the energy (due to this part of the flux) 
428' Pot &' dv. 
Let us consider the energy due to the irregular flux which will 
belong to the above defined element Dv, which is not infinitely 
small, but which has the advantage of being one of physically 
similar elements which make ty the whole medium. The 
energy of this element is found by adding the energies of all 
the infinitesimal elements of which it is composed. Since 
there are quadratic functions of the quantities é, th; C, which are 
sensibly constant throughout the element Dv, the sum will be 
quadratic function of é, ts . say 
(A'S 4B + CS +E e+F'2E+G'S7) Dv, 
which will therefore represent the energy of the element Du 
due to the irregular flux. The coefficients A’, B’, etc., are 
determined by the nature of the medium and the period of os- 
cillation. They will be constant throughout the medium, since 
one element Dv does not differ from another. 
