170 General Analytical Theorems [OH. vn 



and this must shew itself in an equal increase in the electric energy. The 

 electric energy W can be put in either of the forms 



W = W, = 4 fjjpVdxdydz, 



1 /T/V {ft v V fiV\* fiVV] j j j 



TF= TF 2 = -- K \(^-} + U + hr \dxdydz. 

 STrJJJ (\dxj \dy J \OZ J } 



When the displacement takes place, there will be a slight variation in 

 the distribution of electricity and a slight alteration of the potential. 

 There is also a slight change in the value of K at any point owing to 

 the motion of the dielectrics in the field. Thus we can put 



where (STPi) P denotes the change produced in the function W l by the varia- 

 tion of electrical density alone, (TFi) F that produced by the variation of 

 potential alone, and so on. 



We have 



^ 



(o Tr a ) F = A U -^ + ^ ^ -- h ^ -- ^ dxdydz. 

 47rJjJ \dx dx dy dy dz dz J 



By Green's Theorem, the last expression transforms into 



= 1 1 \p$V dxdydz, 



so that 



We accordingly have 



the variation produced by alterations in F no longer appearing. 

 Now (3 W Op = i ! jjfy V dxdydz, 



so that STF= VSp-~SK\dxdydz .................. (113). 



07T J 



