applied to Statical Electricity. 19 



a hypothetically "perfect" solid*, in which 



5m=6fi, (110) 



so that we must use equation (108). 



Prop. XIV. — To correct the equations (9)f of electric currents 

 for the effect due to the elasticity of the medium. 



"We have seen that electromotive force and electric displace- 

 ment are connected by equation (105). Differentiating this 

 equation with respect to t, we find 



_ = _4^_, (m) 



showing that when the electromotive force varies, the electric 

 displacement m also varies. But a variation of displacement is 

 equivalent to a current, and this current must be taken into 

 account in equations (9) and added to r. The three equations 

 then become 



P ~4nr\dy dz W dt)' 



dy dz E 2 dt) 

 q ~~ 4xr\dz dx E 2 dt)' 

 r ~~ 4nr\dx dy W dt)' ■* 



(112) 



substituting, we find 



where p, q, r are the electric currents in the directions of a, y, 

 and z\ u, ft, y are the components of magnetic intensity ; and 

 P, Q, R are the electromotive forces. Now if e be the quantity 

 of free electricity in unit of volume, then the equation of conti- 

 nuity will be 



i+f -4:+!=° • < ii3 > 



Differentiating (112) with respect to.r, y } and z respectively, and 

 ind 



dt ~ 4ttE 2 dt\dx ^ dy^ dz)' * ' K ' 

 whence 



1 /dV , dQ , dU\ ni „ 



e =^\T X + Ty+lz-)' ' ' < 115) 



the constant being omitted, because e — when there are no elec- 

 tromotive forces. 



Prop. XV.- — To find the force acting between two electrified 

 bodies. 



The energy in the medium arising from the electric displace- 



* See Rankine "On Elasticity," Camb. and Dub. Math. Journ. 1851. 



t Phil. Mao-. March 1861. 



C2 



