200 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



hjl y ■ COS {JIkJi r) 

 P = . 



where ^ = the permeability, k,c and hy the gradients of the suscepti- 

 bility and resultant magnetic potential function, respectively, and 

 {Hk, hy) is the angle made by the directions in which k and F increase 

 most rapidly. For we have by Poisson's Equation, 



V-r=-47rp, 

 and from the fundamental equation of magnetic polarization, 



p = _ Divergence / = - [|; (x-V) + |^ (<< F) + ^ (kZ)~^ • 



VdK_ dV dK dV 8k ar"| 



I dx dx dy dy dz dz J 



K'VW + ^. 



Eliminating the v^P^'we get the equation above. Now A^, hy, and jx 

 are all intrinsically positive. The Jik becomes zero under special con- 

 ditions, and is vanishingly small when the iron becomes fully satur- 

 ated. Therefore the sine of o- is governed by the cos (/?«, h^ alone. 

 Considering only the half of the iron cylinder on which the positive o- 

 appears, we see that V always increases from the end of the rod 

 toward the centre, while p does so as long as the magnetization at the 

 centre of the rod has not been pushed beyond the maximum suscepti- 

 bility point. Under these conditions (/?«, hy) is an acute angle, and 

 therefore p is positive. Therefore the argument regarding the curva- 

 ture of the iV-curve in the neighborhood of the origin is even strength- 

 ened all the more on account of the positive p intensifying the 

 demagnetizing force. Thus the lower curvature is proved (although 

 not quite rigorously, mathematically speaking), and since the i\-curve 

 must end in the point K, there must be a curvature in the upper part 

 of the ^-curve directed oppositely to the first one. 



An interesting fact perhaps worth noticing in regard to the volume 

 distribution p of the magnetism is that as soon as the point of maxi- 

 mum susceptibility has been passed over, which will first occur at the 

 centre of the rod, there will appear some negative p near the centre of 

 the rod in that half of the rod which always carries the positive sur- 



