226 MR. G. H. LIVENS ON THE 



another and more involved form. The fundamental basis of this theory is the 

 assumption of a distribution of true magnetic matter of density at any place equal to 



p m = div I = -- div B 

 4rr 



wherein I is the density of the permanent magnetic polarity. This magnetic matter 

 is supposed to be distributed continuously throughout the space but so that the 

 amount in any portion of the matter is zero, a condition which is perhaps rather 

 difficult of realisation, as it would make the distribution in any particular portion of 

 the matter dependent on the distribution in all the surrounding portions. 



In this theory the magnetic energy is first calculated on analogy with the electro- 

 static energy ; the magnetic induction vector B is regarded as a sort of composite 

 displacement produced by the acting force H, so that the energy per unit volume is 



This expression is then verified to be eqtiivalent in the purely statical case to the 

 volume integral 



j dv j <j> dp m 



taken over the entire field, the surface integral over the infinite boundary 

 contributing nothing in all regular cases ; <f> is the magnetic potential of the field. 



In generalising the theory to the case where the field is due to linear currents the 

 same physical basis is adopted as regards the expression 



which still therefore remains valid, and when there are no permanent magnets about 

 this is easily verified by the usual argument to be equivalent to the summation 



1 f N 

 iz 

 c J 



JdN 



over the different current elements, J denoting the typical current strength and N 

 the induction through its circuit. When there are permanent magnets present this 

 expression becomes 



It is then shown that the mechanical forcive on the magnetic matter in any one of 

 its co-ordinates is derivable as the appropriate negative gradient of this energy 



