IN PERFECTLY TRANSPARENT MEDIA. 197 



we may regard or as a quadratic function of (, r\, f and diff. coeff., or 

 as a linear function of the seventy-eight squares and products of these 

 quantities. But the seventy-eight coefficients by which this function 

 is expressed will vary with the position of the element of volume with 

 respect to the surrounding molecules. 



In estimating the statical energy for any considerable space by the 



integral r 



f<rdv, 



it will be allowable to substitute for the seventy-eight coefficients 

 contained implicitly in or their average values throughout the medium. 

 That is, if we write s for a quadratic function of q, f, and diff. coeff. 

 in which the seventy-eight coefficients are the space-averages of those 

 in or, the statical energy of any considerable space may be estimated 



by the integral r 



Jsdv. 



(This will appear most distinctly if we suppose the integration to 

 be first effected for a thin slice of the medium bounded by two 

 wave-planes.) The seventy-eight coefficients of this function 8 are 

 determined solely by the nature of the medium and the period of 

 oscillation. 



We may divide s into three parts, of which the first (s,) contains the 

 squares and products of TJ, f, the second () contains the products of 

 f, *l> f with the differential coefficients, and the third (,) contains the 

 squares and products of the differential coefficients. It is evident that 

 the average statical energy of the whole medium per unit of volume 

 is the space-average of s, and that it will consist of three parts, which 

 are the space-averages of s, , s tt , and 8 HI , respectively. These parts we 

 may call S, , S y/ , and S,,, . Only the first of these was considered in 

 the preceding paper. 



Now the considerations which justify us in neglecting, for an 

 approximate estimate, the terms of s which contain the differential 

 coefficients of ij, f with respect to the coordinates, will apply with 

 especial force to the terms which contain the squares and products of 

 these differential coefficients. Therefore, to carry the approximation 

 one step beyond that of the preceding paper, it will only be necessary 

 to take account of 8, and 8 H and of S, and S y/ . 



7. We may set 



where, for a given medium and light of a given period, A, B, C, E, F, 

 G are constant. 



Since the average values of 



U 90 U U O U 



-r, cos 2 27Ty, sm ZTT j- cos ZTT j 



