192 Mr. G. A. Schott on the 
of the changing field at starting, which is partly external, 
partly due to the changes in radius and velocity of the ring- 
studied in § 18 and represented by equations of the type (19). 
It is not difficult to see that the part of the couple due to 
the ring bears to the part directly due to the external field a 
ratio, which is of the same order of smallness as the ratio 
p x : p 2 in § 20. The corresponding parts of the magnetic 
field due to the sphere bear to each other a ratio of the same 
degree of smallness, and therefore also do the resulting parts 
of the magnetic moment of the sphere. The large part we 
shall find to be of the order p 2i the small part of the order 
p ± ; the latter we shall neglect. 
For the same reason we neglect the moment p 4 , and the 
, change in it due to the reaction of the sphere ; and, in 
estimating the change produced in p 2 by the sphere, we take 
into account only the rotation due to the external field. This 
we shall now consider. 
Let L be the resulting couple, and fl the corresponding- 
angular velocity ; then 
4- J 
where I is the moment of inertia*of the sphere. 
Divide the sphere into elementary rings with the diameter 
parallel to the external magnetic force h as axis ; let d V be 
the volume of a ring, p' its radius, e the uniform electric 
density, and e' the charge of the sphere. The induced 
electric force E acts along the ring and is by Faraday's Law 
given by 2tt/>'E= — - T (*ny>' 2 /i), so that E = - £- lu Hence 
C itt £C 
also 
J 2c DC 
and ~ e'lr . ^^ 
n= -57i h ' ( a2 > 
the sphere being supposed at rest in the absence of an 
external field. 
§ 24. We shall now study the field due to the sphere. In 
the first place, the external field is the same as that due to an 
infinitely small magnet at the centre, whose moment is 
given by 
e'V- e%\ 
