472 J. W. Gibbs— Elastic and Electrical Theories of Light. 



of the constant G, which was sufficient when the ponderable 

 particles were absent. Both G D and £ D will vary to some ex- 

 tent with the period, like A D and/^, and for the same reason. 

 In regard to that potential energy, which on the elastic 

 theory is independent of the direct action of the ponderable 

 molecules, it has been supposed that in seolotropic bodies the 

 effect of the molecules is such as to produce an seolotropic state 

 in the ether, so that the energy of a distortion varies with its 

 orientation. This part of the potential energy will then be 



represented by B ND ^ , where B NJ ; is a function of the directions 



of the wave-normal and the displacement. It may easily be 

 shown that it is a quadratic function both of the direction- 

 cosines of the wave-normal and of those of the displacement. 

 Also, that if the ether in the body when undisturbed is not 

 in a state of stress due to forces at the surface of the body, 

 or if its stress is uniform in all directions, like a hydrostatic 

 pressure, the function B ND must be symmetrical with respect 

 to the two sets of direction-cosines. 



The equation of energies for the elastic theory is therefore 



A D - = B ND -= + b D h\ (5) 



p i 



which gives 



~ p 2 A D —b D p 2 ' { ' 



The equation of energies for the electrical theory is 



F^ +f„p = G B h\ (7) 



which gives 



v - f ~ f ~ iy- (8) 



It is evident at once that the electrical theory gives exactly 

 the form that we want. For any constant period the square 

 of the wave-velocity is a quadratic function of the direction - 

 cosines of the displacement. When the period varies, this 

 function varies, the different coefficients in the function vary- 

 ing separately, because G D and f^ will not in general be simi- 

 lar functions.* If we consider a constant direction of displace- 

 ment while the period varies, G D and f^ will only vary so far 

 as the type of the motion varies, i. e., so far as the manner in 

 which the flux distributes itself among the ponderable mole- 



* But G D , / D , and V 2 , considered as functions of the direction of displacement, 

 are all subject to any law of symmetry which may belong to the structure of the 

 body considered. The resulting optical characteristics of the different crystallo- 

 graphic systems are given in volume xxiii of this Journal, page 273. 



