( G26 ) 



revoliilioii (»r S ahoiil llio axis /*Q is iiiimci'icallv o(|iiaI lo llio pivxliicl of 



t 



— \)\ tlic iiiiiiiIk'I- (»r lilies of (orcc' llial are ciil l)v S. 'Hicsc iii)(^s 

 c 



are |»r('cis('lv lliosc (lial ai'c iiilci'scckMl hv llic siii-racc (Icscrihcd i>v 

 *S' in ils rcvoliilioii, a siii'faco wliicli inay liav(i (li/ïereiit lornis, accor- 

 ding- to llic form of llic wire 11^ l)nl lias at all events for its houn- 

 (lai'ies tlie circles desci'ihed hy llie points (/ and //. Let iV he tiie 

 niiniher of' tlicsc lines, laken j)Ositive illlie iiiiddk; oiu? of" llieiii passes 

 'tijtinn-ds aloii';- PQ, and lel ns lake as positive directions for the 

 rolalion and foi' the couple the dii-ection correspoiidiiir^ to llie nj)vvard 

 dii-ection. Tlieji, for a full revolution in the positive direction, the 



work of the couple \v\\] bo ■ I lY, whence we find for fluM-onplc 



c 



itself 



-^i^- . (24) 



2 TIC 



If this were all, we should indeed c(jnie to an eil'ect such as was 

 expected by Whitkhkad. We mnat however keep in mind that tiierc 

 can never be a variable magnetic Held without electric foi-ces. Such 

 forces, represented in direction and inlensily by the vector b, will 

 exist in the field F^, the lines of electric foi'ce being circles around 

 the axis PQ. 



We niusl Iherefore add to (24) the (•ouj)le arisijig from the action 

 of the field on the charges e and — e; its moment may again be 

 found by considei'ing the work done in a complete revolution in the 

 positive direction. 



The foi'ce on the charge e being r b, its work is equal to the 

 product of e by the line-integral of b along the circle described by 

 (r. Similarly, the work of the force acting on the charge — e in N 

 is the pi-oducl of — e by the line-integral of b along the circle 

 descrii>ed by //, or, what amoiiiils lo the same thing, the product 

 of 4~ '' '\V ^''*^ line-integral for this circle, if it is taken in the 

 negative direction. Now, if we follow the cii'cle (i iji the positive 

 and the circle H in the negative direction, we shall have gone along 

 the whole contour of the surface described l)y Ihe stream-tube ^S', 

 in a direction corresponding to the positive direction of the magnetic 

 force. Hence, by a well known theorem, of which the fundamental 

 equation (VI) is the expression, the sum of the two line-integrals l)y 

 which e must be multiplied, will be 



e (IN 

 c dt 



