200 >F THE AMEBIOAS ACADEMY. 



hJl T > COS (k K ,h 



P = 'I 



where u = the permeability, lu and //,- the gradients of the 3U8oepti 

 bility and resultant magnetic potential function, respectively, and 

 (//... />■) is the angle made by the directions in which k and V u 

 rapidly. Fur we have by Poisson's Equation, 



V-r = -47rp, 

 and from the fundamental equation of magnetic polarization, 



— I 



— (kA") + (kY)+ -(kZ 



= B ft l'\ ft V\ r fd_V\ 



dx\ (■'■ J > </\ ',, ) cz \ c: J 



Eliminating the v 2 J'we get the equation above. Now h K , //,-, and » 

 are all intrinsically positive. The h K becomes zero under Bpecia] con- 

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

 ated. Therefore the sine of o- is governed by the coa (h*, hf) alone 

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

 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- 

 hility point. Under these conditions (h K) hr) is an acute angle, and 

 therefore p is positive. Therefore the argument regarding the curva 

 ture of the .V-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 A^curve 

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

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



An interesting fact perhaps worth noticing in regard to the volume 

 distribution p of the magnetism is thai as si as the point of maxi- 

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

 ire 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- 



