STUDIES ON MAGNETIC DISTRIBUTION 91 



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 experi- 

 ments of Miiller, and more recently by those of Dr. Stoletow and my- 

 self, 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 electricity 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 n 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, 



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



s be the resistance at the end of the rod, 



Q' 4 be the number of lines of induction passing along the rod 



at a given point, 

 $'. 5 be the number of lines of induction passing from the rod 



into the medium along a small length of the rod JL, 

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



R ' 



A _ V RR' + s 



, dL 



+ dp= ,57 



To find ft, the ordinary equation for the resistance of a derived cir- 

 cuit gives 



whence 



4 These are the surf ace-integrals of magnetic induction (see Maxwell's ' Electricity,' 



art. 402) the first across the section of the bar, and the second along a length AZ, 



of the surface of the bar. 



5 It is to be noted that Q', when A is constant, is nearly proportional to the so- 

 called surface-density of magnetism at the given point. 



