and the Maximum of Magnetism of Iron, Steel, and Nickel. 145 

 <cb, and £=x , we have 



Q. = 0, and Q'= J ...... (3) 



For a ring-magnet ; 6 7 = j 



.-. Q,=0, and Q'= ^ (4) 



And if a is the area of the bar or ring, 



> i Q' „ Q' 



aX= E = M' 0l ' X= ^M' • • • • P) 



in which X is the same as in the equations previously given. 

 These equations show that we may find the value of X, and 

 hence the permeability, by experimenting either on an infi- 

 nitely long bar or on a ring-magnet. Equations (4) evidently 

 apply to the case where the diameter of the ring is large as 

 compared with its section. The fact given by these equations 

 can be demonstrated in another and, to some persons, more 

 satisfactory manner. If n is the number of coils per metre 

 of helix and ri the number on a ring-magnet, i the strength of 

 current, and p the distance from the axis of the ring to a given 

 point in the interior of the ring-solenoid^ the magnetic field 

 at that point will, as is well known, be 



2n'i -, 

 P 



and at a point within an infinitely long solenoid 



47rm. 



If the solenoid contain any magnetic material, the field will 

 be for the ring 



2n'i^ 

 P 



and for the infinite solenoid 



4arfiin. 



Therefore the number of lines of force in the whole section of 

 a ring-magnet of circular section will be, if a is the mean radius 

 of the ring, 





— — dx = 47rn'i/JL(a — \/a 2 — R 2 ) ; 



or, since n'^Ziran and M = m, we have, by developing, 



Q'=47rM^R*)(l+ig + ^-<-&c.). . . (6) 



