Ox THE MAGNETIC EFFECT OF ELECTEIC CONVECTION 133 



fraction of a per cent, we may consider the two sides of the disc to 

 coincide in the centre. Taking the origin of coordinates at the point 

 of the disc under the needle and the centre of the disc on the axis of X. 

 we find for both sides of the disc, the radial component of the force 

 parallel to the disc, 



r c ~ f 

 J_ (C+b) J. 



x)dxdy 



(a 1 + a? + 



f> - (b 



where a is the distance of the needle from the disc and & that from 

 the axis; N is the number of revolutions of the disc per second and 

 v = 28,800,000,000 centimetres per second according to Maxwell's de- 

 termination. The above integral can be obtained exactly by elliptic 

 integrals, but as it introduces a great variety of complete and incom- 

 plete elliptic integrals of all three orders, we shall do best by expanding 

 as follows: 



V 4-JW 7, faNff f . . A a >. -r.v 



X= - P - (A! + A* + A 3 + &c.), ... (4) 



A, = 2jfarc tan -=^ + arc tan ^-^ - a log, 4 , 

 \ a a ] JV 



2sb + a2) loge 



(5s 3 



&c., &c., 

 where 



-, , . 



/it) 



From this must be subtracted the effect of the opening in the centre, 

 for which the same formula will apply. 



The magnetic action of the excess at the edge may be calculated on 

 the supposition that that excess is concentrated in a circle of a little 

 smaller diameter, C", than the disc; therefore, 



