466 J. W. Gibbs—Double Refraction and Circular 
radius is small in compariehs with a wave-length,* and since 
within such a sphere é, ’; ¢ and diff. coéf. are sufficiently de- 
termined (in a linear form), by the values of the same wales 
quantities at the center of the sphere, it follows that Eat &, 
Pot 7, Pot e must be linear functions of the values of &, 1'; 4 
and diff. coeff. at the point for which the potential is sought. 
Hence, 
4(&' Pot 2’ +1 Pot 7 +2 Pot 2’) 
will be a quadratic function Dae a i, C e and dif. coef. But the 
seventy-eight coéfficients this function is expressed 
will vary with the position of the int considered with respect 
to the surrounding molecules. 
Yet, as in the case of the statical energy, we may substitute 
the average values of these coéfficients for the coéfficients 
themselves in the integral by which we obtain the energy of 
any considerable space. The kinetic energy due to the irregu- 
lar ‘part of the flux is thus reduced to a quadratic function of 
é, ” z and diff. coéff. which has comet coéfficients for a 
given medium and light of a given 
The function may be divided tno. tie parts, of which the 
first contains the squares and produets of &, ” C, the second the 
eae pa of &, qs ¢ with their differential coGfficients, and the 
, which may be neglected, the squares and products of the 
differential coéfiicients 
e may proceed with the reduction precisely as in the case 
of the statical energy, except that the differentiations with re- 
x 
spect to the time will introduce the constant factor —-. This 
will give for the first part of the kinetic energy of the irregular 
ux per unit of volume 
T= (Was + Bis +Cy +EBy, +F'y,a,+G'a,f,) 
+ 2B (Aas + BAS + Cy! +E By.+P'y.a,+@ah,), (1) 
and for the second part of the same 
atte 
= PTL By. 1 B)+Mya,—ay)+N(af,—Ba)h (12) 
* See § 9 of the former paper, on page 268 of this volume. 
