266 Mr. J. Parker on the Theory of Magnetism and 



the contents of the cylinder before removal, and V the poten- 

 tial of the whole of the given system, we should have 



and therefore 



dx dx dx ' *' 



Now the contents of the small cylinder, before being cut 

 away, were magnetically equivalent to layers on the ends, of 

 densities +1 on the positive end and —I on the negative 



end. Thus -. — is simply the force, parallel to the axis 



of x, arising from these two layers. But if we take a cir- 

 cular layer of uniform density I, the force it exerts on a 

 unit pole in the axis of the layer at a point where the radius 

 of the layer subtends an angle a, is 27rl(l— cos a), and may 

 therefore be neglected when a is small. Hence, if the radius 

 of the right circular cylinder be infinitely small in comparison 

 with the length, the differential coefficients of Y" will be zero. 



Consequently, ( — ~j~>"~~7~ ?""~"T~) are ^ ne ma g n etic forces 



parallel to the axes, exerted by the new system V on a unit 

 pole placed at P without disturbing that system. These 

 forces are written (X, Y, Z) , and are called the forces of the 

 given system at P. 



If F be the resultant of (X, Y, Z), or the resultant force 

 of the given system at P, equations (4) become 



X_.Y Z F l tritr . 



A-B = G- ± T = r^ (I '^ 



Now it has been shown by Duhem (Des Corps Diamag- 

 ne'tiques) that i|r(I, 6) must always be positive. We must 



TCI 



therefore always take the positive sign before y, and may 



write 



! = T = E=ff=x(I,*), • • • • ( 5 ) 



where %(I, 0) is always positive. 



The meaning of equations (5) is that, at any point of a 

 " perfectly soft " homogeneous substance, the magnetization, 

 when in stable equilibrium, coincides in direction with the 

 force at that point. If there is any magnetic " rigidity " 

 about the substance, the magnetization at a point may, of 

 course, make a finite angle with the force at that point. 



