428 Mr. A. Russell on the Magnetic 



magnetic force at an element r r d6 f dr f of the area of the circle 

 whose radius is b, then 



"' ^" LW 7 



OjO 



i 



and noticing that r*=r r2 + c 2 , and /cos# = c, we get 

 M _C b C" _ (a-r f cos <\>)r'dO f dr' 

 47ja ~" JoJo (cf+ r /2 + c'-2ar' cos cpf! 2 ' 

 Changing to rectangular coordinates, we have 



l/{(«-^)2 +3/ 2 + c 2 } l/2l dym 



'0 L J _ ^52-5,2 



Put y = b sin and simplify. In the second integral also, 

 write <^)' = 7r — <fi. We get 



M _ p 7 cos </x/0 



Now putting (p = 7r — 26 J we have 



47T«6 rj A 



and therefore, by (4) 



M=4w\/S{(2/ife-*)F-(2/i)E}, . . (20) 

 where ^ = (a + J)2 + ^ and p = 4^2. 



Formula (20) agrees with that given by Helmholtz * for 

 the analogous problem of the vortex ring. 



If we transform (20) by the Landen-Legendre formulae (9), 

 we get 



M=^#(F'-E'), .... (21) 

 where 



)t' = (r 1 -r 2 )/(r 1 + r 2 ), rf = (a 4 b) 2 + c 2 , and v? = (a - 6) 2 + c 2 . 

 This is Maxwell's formula. 



iii. Approximate formula for the self -inductance of a ring. 



We shall suppose that the radius of the ring is a } that the 

 radius of the cross section is r, and that the current density 

 at all points of the cross section is the same. We suppose 



Crelle's Jourmd, vol. lv. p. 25 (1858), or Ges. Abh. t. 1, p. 101. 



