of Force between Electric Currents. 463 



that the vector potential stands in exactly the same relation 

 to the elementary current in which the potential of a particle 

 of matter stands to that particle — a statement which leads to 

 all the consequences we now point out. 



19. Again, the mutual potential energy of any two elemen- 



COS G 



tary currents ids, i' ' ds r being ^w'— - dsds', may be put in 

 the form r 



. dx . dy , dz 



f ds ,,da/ ds ..dif , ds ..dz' \ . 7 , 



*■ r ds' r ds r ds' ) 



. dx 

 Now i -j- ds is the component part of the current i ds in the 



direction of x. If we assume, as we must, that the current, 

 instead of flowing in an infinitely thin line, is (like a fluid of 

 finite density) distributed uniformly over a small section per- 

 pendicular to ds, the x component will in like manner be dis- 

 tributed uniformly over the element of area dy dz. If we now 



• d tc dz/ d z 



write u, v, to for i — ; i -f-, i — , when therefore u, v, to are the 



CIS CIS CCS 



velocities of the current at ds parallel to the coordinate axes, and 



dx f 

 in like manner u' , v r , w f for %' — &c, we shall have for the 



O/S 



mutual potential energy of the two elements the expression 

 fi(-u f '{--v , + —w' \ dx dy dz dx r dy' dz'. 



A similar expression holds for the mutual potential of every 

 pair of elementary currents. We may therefore express the 

 whole potential of any system of currents in the form of a 

 sextuple integral 



2T=M ffffIf{ TO/+ t +w } * * dz dx ' #"> 



or if 



(TO;' dx'dy'dz' = F, 



$^clv'dy'dz' = G, 

 $^dx>dy>ch'=B, 



2T=fiU\(Fii + Gv + Kiv)dxdydz. 

 2L2 



