ON PHYSICAL LINES OF FORCE. 497 



Differentiating (112) with respect to x, y, and z respectively, and substituting, 



we find 



de 1 d IdP , dQ , dR\ 



dt kirE* dt \dx dy dz 

 1 IdP , dQ , 



the constant being omitted, because e = Q when there are no electromotive forces. 



PROP. XV. To find the force acting between two electrified bodies. 

 The energy in the medium arising from the electric displacements is 



U=-^(Pf+Qg + Rh}W. .................... (116), 



/ 

 where P, Q, R are the forces, and /, g, h the displacements. Now when there 



is no motion of the bodies or alteration of forces, it appears from equations (77)* 

 that 



*--.<?--.*-- .................. < 



and we know by (105) that 



(119); 



whence *- + + * ................... ( 120 )' 



Integrating by parts throughout all space, and remembering that "9 vanishes at 

 an infinite distance, 



(121); 



or by (115), U=&(Ve)8V .............................. (122). 



Now let there be two electrified bodies, and let e 1 be the distribution of 

 electricity in the first, and j the electric tension due to it, and let 



1 1*9 d^ d^\ 

 + 3 ^ ~ 



Let ei be the distribution of electricity in the second body, and W a the 

 tension due to it ; then the whole tension at any point will be % + and 

 the expansion for U will become 



U=& (*& + *& + *& + *&) 8V. .................... (124). 



* Phil. Mag. May, 1861 [p. 482 of this vol.]. 

 VOL. I. 63 



