188 DOUBLE REFRACTION AND THE DISPERSION OF 



indicates that we may regard the infinitesimal element dv as having 

 the energy (due to this part of the flux) 



JSf Potf 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 up 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 these are quadratic functions of the 

 quantities , r\, f, which are sensibly constant throughout the element 

 Dv, the sum will be a quadratic function of r\> say 



v, 



which will therefore represent the energy of the element Dv due to 

 the irregular flux. The coefficients A', B', etc., are determined by the 

 nature of the medium and the period of oscillation. They will be 

 constant throughout the medium, since one element Dv does not differ 

 from another. 



This expression reduces by equations (4) to 



i'a 2 + B'/3 2 + C'y 2 + E'/3y + F'ya + G'a/3) cos 2 ZTT ~ D v. 



JJ i 



The kinetic energy of the irregular flux in a unit of volume is there- 

 fore 9_2 



T = ^- (A'a 2 + B^ 2 + C'y 2 + E^y + Fya + G'a/3). (9) 



10. Equating the statical and kinetic energies, we have 



ya + G / a ) 8). (10) 

 P P 



The velocity (V) of the corresponding system of progressive waves is 

 given by the equation 



2 _ 1 

 ~~ 



27T AV + B'/^ + Cy + EV3y + FVa + G / a / 3 



O 9 i /^O i O * \ / 



p* a^ + p^ + y^ 



If we set 



S-^B', etc., (li 



/v\Z y ' ^ 



and 



the equation reduces to 



