( 6(5 ) 



O 



Wlial lias alreadv heoii said about \ho soUMioidal distrihiilioii of 

 dl is coiifii-UR'd hv lliis ('(|iialioii. Tlic Iwo vcciors iv|)rescMiled on llic 

 ri,i>,-lil hand side holli lia\(' lliis pi-opiM'lv, die lirsl l)_v wliat we know 

 of the vectoi- ff^-[-o.1. and llie second on acconnl of the niatliema- 

 tical foi'ni in \\ hicli it ai)})ears. 



§ (i. We mav ne.\( proceed to delerniine tlie variation ff7'oflIie 

 niajiiietic energv. In doin,^- so we shall start from the assnniption 

 thai the vai-ied motion of electricity iinolves a detinite magnetic 

 energy ^), to he delermined as stated at the end of § 1. 



The formula 



1 r 



leads immediately to 



frr = j{iy, rff)., -f- b, ff:,,-}- f), rff),) dS = f{l) . fff)) .1^, 



where the integration covei-s all s])ace. The same will he tlie case 

 with the other xolume-integrals ap|)earing in the following transform- 

 ations. If aji integration is i)erfoi'med, or if the jinx'ess of inte- 

 gration hy [)arts is applied, one ohiains ijitegrals over the ijitinite 

 surface which we may concei\e as the houndai-y of the tield of inte- 

 gration. These surface-integi-als however will he supposed to vanish. 

 We begin I>y writing rof a instead of \\ as may he don(^ in A'irtue 

 of (5) ; and we shall next integrate hy pai'ts, lvee])ing in mind that, 

 on account of *(V), 



1 

 rof. ÖI) r= — ffl. 



(.; 



The result is 



(fr =: I (vol a . (f{)) 'AS' = I (a . };>t <i{)) </S = - I [a . <U) dS , . (9) 



or, if we substitute foi- <fl its value (8), 



'^^^ vjh''' ^I'^^^^'^M)'^-'^-^ yrra.>r,/L|.-i.i>]jys. (10) 



Losing (4), we may |»ut for the lii-st tei-m 



\) This assumption only niouns to (ji^tino tlio value of T we shall assign to the 

 wholly fictitious varied slate. 



