190 PROCEEDINGS OF THE AMERICAN ACADEMY. 



in, we shall hereafter use the scalar values for H', Hi, and /, so that 

 our first equation will become 



H = H' -Hi = H' - NI. 



The only case of a magnetized body not endless, in which we can 

 always calculate what the Hi will be,' is where an iron ellipsoid is 

 placed with one of its axes parallel to a uniform magnetizing field H. 

 If the equation of the ellipsoid is 



1- — -^ — =1 



a^ b- c^ 



then it is shown in text-books on the mathematical theory of electric- 

 ity and magnetism,^ that if there exists on the ellipsoid a surface dis- 

 tribution of magnetic matter everywhere equal to 



o- = /• cos {x, n) 



where / is a constant, and (x, n) is the angle between the positive 

 direction of the .^--axis and the exterior normal to the ellipsoid, the 

 volume density p being zero throughout the ellipsoid, then the mag- 

 netic field due to this distribution is constant at every point within 

 the ellipsoid and equal to 



where Kq = I 



^ 



Hi = 2TrabcIKQ, 

 ds 



(s + a)\s + b)^(s + c)^ 



This field Hi is directed parallel to the negative direction of the ^--axis, 

 and tends to demagnetize the iron ; we see furthermore that it is di- 

 rectly proportional to /. The constant / is simply the intensity of 

 magnetization, uniform within the ellipsoid. To keep this magnetic dis- 

 tribution in equilibrium it is sufficient if we apply a uniform magnetic 

 field parallel to the positive .r-axis, of such a strength H', that when 

 diminished by the demagnetizing field Hi, there will remain in the 

 ellipsoid the uniform resultant field H=I/k, where k is the suscepti- 

 bility corresponding to the magnetization /, for the kind of iron under 

 consideration. Of course if the o- has initially been chosen greater 

 than the maximum value of magnetic intensity attainable, it will be 



8 Max\Yell, II, §§ 437 and 438 ; Webster, Elec. and Mag., §§ 192, 196 ; Peirce, 

 Newtonian Potential Function, § 69. 



