198 DOUBLE REFRACTION AND CIRCULAR POLARIZATION 

 are respectively J, J, and 0, and since at the time to be considered 



t 



COS 2 27T 



p 



1, 



it will appear from inspection of equations (1) that 

 S, = K AV + B& 2 + C yi 2 



(6) 



This is the first part of the statical energy of the whole medium 

 per unit of volume. 



8. The second part of the statical energy of the whole medium 

 per unit of volume (S,,) is the space-average of s /x , which is a linear 

 function of the twenty-seven products of , tj, f with their differential 

 coefficients with respect to the coordinates. Now since 



dx 



dx ' dx 



dx ' 



the space-average of such products will be zero, and they will 

 contribute nothing to the value of S x/ . There will be nine of these 

 products, in which the same component of displacement appears 

 twice. The remaining eighteen products may be divided into pairs 

 according to the letters which they contain, as 



dt , ,dn 

 n -5*- and c -r. 



dx * dx 



A linear function of the eighteen products may also be regarded 

 as a linear function of the sums and differences of the products in 



such pairs. But since 



d ,fdrj_ 

 dx ^ dx~ dx 



the terms of a ft containing such sums will contribute nothing to the 

 value of S, . We have left a linear function of the nine differences 



dx ^ dx' ^ dx ^ dx' * dx dx' 



(the unwritten expressions being obtained by substituting in the 

 denominators dy and dz for dx), which constitutes the part of s tl 

 that we have to consider. S y/ is therefore a linear function of the 

 space-averages of these nine quantities. But by (3) 



_ L 

 -^ 



and the space-average of this, at a moment of maximum displacement, 

 is bv ( 1 ) 2?rL 



