and the Partition of Energy in Continuous Media. 2^1 



If we introduce new quantities a, j3, y, X, Y, Z, associated 

 with each cell and given by 



Kpqr = Zpqr 

 ^pqr = -r \ £p,q+\,r—~Zp,q,r Vp,q,r+l + Vp,q, f> • • \P^J 



&c, then the energy-function becomes 



E=^222(* 2 + /3 2 + 7 2 + X 2 + Y 2 + Z 2 ). . (31) 



o7T p q r 



If we make I and o> vanish in the limit, the value of E 

 becomes 



E = ~ |W (a 2 + /3 2 + r + X 2 + Y 2 + Z 2 ) ^ dy dz, 



which is identical with the electromagnetic energy in free 

 aether. The equations of motion obtained from equation (29) 

 reduce, when I is made to vanish, to the equations 



1 dX By 

 C dt ~'d;j~ 



9/3 

 "3* 



1 da. ^Z 



C dt ~~b[i~ 





which are the electromagnetic equations in free aether. 



Thus the mechanical system now under discussion becomes 

 identical dynamically with the electromagnetic field when 

 1 = 0. Any dynamical property of the present system which 

 is independent of / must accordingly be a property of the 

 electromagnetic field. 



The energy function (29) is the three-dimensional analogue 

 of the one-dimensional energy-function given by equations 

 (10) and (11). It can accordingly be expressed as the energy 

 of trains of waves, following the method of § 17. From 

 this it follows that in the " normal state " the law of partition 

 of energy between waves of different wave-lengths must be 



8irRTX-*d\ (32) 



so long as A, is large in comparison with /. Hence this must 

 be the law of partition of energy in the " normal state " in 

 an electromagnetic field, at any rate for waves which are long- 

 in comparison with the scale of structure (if any) of the 

 aether. 



S2 



