Mr. H. A. Rowland on Magnetic Distribution. 259 



of magnetic potential of those points divided by the resistance 

 to the lines. 



The complete solution of the problem before us being impos- 

 sible, let us limit it by two hypotheses. First, let us assume that 

 the permeability of the bar is a constant quantity; and secondly, 

 that the resistance to the liues of induction is composed of two 

 parts, the first being that of the bar, and the second that of esca- 

 ping from the bar into the medium — and that the latter is the 

 same at every part of the bar. The first of these assumptions is 

 the one usually made in the mathematical theory of magnetic 

 induction ; but, as has been shown by the experiments of MUller, 

 and more recently by those of Dr. Stoletow and myself, this is 

 not true ; and we shall see this when we come to compare the 

 formula with experiment. The second assumption is more exact 

 than the first for all portions of the bar except the ends. 



Let us first take the case of a rod of iron with a short helix 

 placed on any portion of it, through which a current of electri- 

 city is sent. The lines of magnetic induction stream down the 

 bar on either side : at every point of the bar two paths are open 

 to them, either to pass further down the rod, or to pass out into 

 the air. We can then apply the ordinary equations for a derived 

 circuit in electricity to this case. 



Let //, be the magnetic permeability of the iron, 



R be the resistance of unit of length of the rod, 



R' be the resistance of medium along unit of length of rod, 



p be the resistance at a given point to passing down the rod, 



s be the resistance at the end of the rod, 



Q'* be the number of lines of induction passing along the 

 rod at a given point, 



Q' e *t be the number of lines of induction passing from 

 the rod into the medium along a small length of the 

 rod AL, 



L be the distance from the end of the rod to a given point, 



a= Vrr/+s 



v/RR'-s' 



To find p, the ordinary equation for the resistance of a derived 

 circuit gives 



* These are the surface-integrals of magnetic induction (See Maxwell's 

 ' Electricity,' art. 402) — the first across the section of the bar, and the 

 second along a length AL of the surface of the bar. 



t It is to be noted that Q' e , when AL is constant, is nearly proportional 

 to the so-called surface-density of magnetism at the given point. 



S2 



