in Conducting Sheets and Solid Bodies. 15 



the mean value of which 



P; 



7ra 3 a> 2 



. 4R 

 and the couple 



_ itc?co sin 2 <f> ™ 



" 2R ,JJ " 



the mean value of which 



7ra B <D p 



4R 

 The ratio of this effect to that produced in the shell is ~ £- 9 



which is small for slow speeds, but increases without limit as 

 the speed increases.] 



We notice that if p is small, 



P 1 — A cos s(f> — A p sin scj> ; 



so that the effect is (1) to neutralize the outside field through- 

 out the interior of the shell, (2) to add on a new weak field 

 equal to the former turned through an angle 7r/2s in the nega- 

 tive direction and multiplied by p. 



For moderate values of s, since R is large, p can be small 

 only by a and co or both becoming large. In particular, for a 

 copper shell -J centim. thick, R=3 x 1640 about: thus, if in a 

 uniform field of force (*=1, n = l)p is to be so small as ^, we 

 must have 



ao) = 5870, 



which it is impossible to realize. In a possible case, say 

 ft) = 27rx 20 (which corresponds to 1200 turns per minute, an 

 ordinary speed for the armature of a dynamo), a=10 centim., 

 thickness of shell =^ centim., we find p = l nearly; and the 

 result is that inside the shell half of the field is rotated through 

 a right angle, the other half remaining as before. If the 

 thickness were 1 centim., we would have _p = 3, the field inside 

 would be diminished to j 1 ^, and a new field of -^ of the former 

 turned through a right angle would be added on. 



(2) For the case of a thick shell or a solid sphere, the equa- 

 tion for P which, when the sphere is at rest, has been found 

 to be 



now becomes 



^ p =va< p+p °)' 



v,p =v'°5? (P+p <' ) (im 



