OF ELECTRO-DYNAMIC FORCES. 523 



Moreover, when, in addition to the motions of the electric masses 

 in their conductors, the motion common to them and their con- 

 ductors is taken into account, we have the following expressions 

 for the first differential coefficients : — 



dr, 



= — u cos + «' cos ®' + V cos »), 



Hence 



dt 



dr, 

 ~di 



dr.. 

 ~di 



dr^ 

 ~dl 



d T 



-rj = + w cos % — ijI cos 0' + V cos tj, 



-^ = — u cos © — m' cos <&' + V cos tj, 



— -^ = + J/ cos © 4- w' cos 0' + W cos )). 



(dr^ _ dri dri dr^\ o , ^ ^, 



V7F + -rf- rfF-^j = -8"«'cos©cos©', 



(dr,^ dr.^ dri ^^/\ 



We obtain the second differential coefficients when the varia- 

 bility of the velocity u, u', and v is also taken into account : — 



t-i.du , _,, du' dv 



-C0S©-^ + C0S©'-^+ COS),^, 



dt dt ' dt 



<a ofM -., du' dv 



+ COS © -T-T - COS©' -— - + COS .] ~, 



dt dt ' dt 



—rr-— + ttsm©-rr + w'sm©'-j— 2 — vsmw-r^ 

 "'s dt dt ' dt 



„ du _., du' dv 



- cos ©-— - COS©' -r- + COST) -yL 



at dt dt 



-j^ = - usm® -jf -u' sm& -^ -vsmri-J>± 



^du , _^, du' dv 



+ cos © -T- + cos©' -— - + COS .) V-. 



dt dt dt 



2 N 2 



