20 Prof. Maxwell on the Theory of Molecular Vortices 



mcnta is 



U=-^(iy+Q,7-fR/0SV, . . . (116) 



where P, Q, 11 arc 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 



P ** Q--^ R--^- (118) 



r ~ dx' U " dy' ii ~ dz' ' [lL * } 



and we know by (105) that 



p = _4 (7r E 2 /, Q=-47rE 2 ^ R=-4mWh; . (119) 

 whence 



~ ,2 '7at>\2 



1 fdWV* (l^r\- dw\ 2 \ 



dy 



Integrating by parts throughout all space, and remembering that 

 M* vanishes at an infinite distance, 



TT l VMr( d ^ d ^ ■ ^W /ion 



or by (115), 



U=i2(^)SV (122) 



"Now let there be two electrified bodies, and let e Y be the distri- 

 bution of electricity in the first, and "f , the electric tension due 

 to it, and let 



^(d^ , d?% , d^ x 



dy* 



Let e 2 be the distribution of electricity in the second body, and 

 X P 2 the tension due to it ; then the whole tension at any point 

 will be M^ -f- ^2, and the expansion for U will become 



U=i2(^ 1 +^ 2 +^ 2 +^ 1 )SV. . . (124) 



Let the body whose electricity is e x be moved in any way, 

 the electricity moving along with the body, then since the dis- 

 tribution of tension M/j moves with the body, the value of '^ ] e l 

 remains the same. 



M^g also remains the same ; and Green has shown (Essay on 

 Electricity, p. 10) that ^$ r 1 e 2 =' , I r 2 e 1 , so that the work done by 

 moving the body against electric forces 



W=SU=8S(^' 2 (? 1 )SV (125) 



And if e x is confined to a small body, 



W=e^ 2 , 





* Phil. Mag. May 186*1, 



