Magnetic Circuits. 337 



KircKhoff then introduced a new theory (in 1853) by making 

 the more general assumption that magnetization is a function 

 (capable of experimental determination) of the total magnetic 

 intensity. Besides the important solutions of certain par- 

 ticular cases (ellipsoid, closed ring), he restricted himself to 

 giviDg a few integral equations in the place of Poisson's. 

 However, these were of but little practical use, and are hardly 

 more so now that Duhem has lately taken up the analytical 

 problem anew. 



A more geometrical treatment of the new theory showed 

 the distributions of the vectors concerned to be as follows : — 

 (a) magnetic intensity : lamellar; (b) magnetization : complex- 

 lamellar ; (c) induction : solenoidal ; {cl) also the three vectors 

 are easily seen to have the same direction at every point. The 

 proof and further discussion of these propositions cannot be 

 given here. It may be remarked that homogeneous, isotropic 

 ferromagnetic substance is assumed, through which no electric 

 currents are supposed to flow ; neither is hysteresis taken into 

 account. 



The simple type of a circuit not completely closed is a thin 

 ring containing a radial air-gap and subjected to a uniform 

 tangential magnetizing force. This particular case, a solution 

 of which has to my knowledge never been attempted, is re- 

 ducible to the known case of an ellipsoid of revolution. In 

 fact it is only necessary that the " self-demagnetizing factor " 

 be capable of calculation ; i. e. the number into which the 

 magnetization has to be multiplied in order to obtain the 

 intensity of the self-demagnetizing effect. Let this numerical 

 factor be, as usual, denoted by N ; for sufficiently long ovoids 

 (prolate ellipsoids of revolution) of axial ratio m it is found 

 by the well-known equation 



N=g(log e 2m-l). 



Now let the gap of our ring have the angular value a ; i. e. 

 an arbitrary concentric circle in it being considered, the 

 a/360 part of this will lie in air, the (360 — a)/360 part in the 

 ferromagnetic substance. A consideration of the line-integral 

 of the self-demagnetizing intensity, which must vanish along 

 any such closed circular line of integration, leads as a first 

 approximation to the result that 



The proof cannot well be given without a diagram. Both 

 particular cases are now comparable in every respect, as the 

 following short table shows : — 



