116 THE MAGNETIC CIRCUIT [ART. 41 



mating the leakage factor, the reader has seen the difficulties 

 involved in the computation of the permeance of an irregular path. 



In the parts of a magnetic field not occupied by the exciting 

 windings, the general principle applies that the lines of force and 

 the equipotential surfaces assume such shapes and directions that 

 the total permeance becomes a maximum, or the reluctance a 

 minimum. When this condition is fulfilled, the energy of the 

 magnetic field becomes a maximum, as is explained hi Art. 57. 



When the field needs to be considered in two dimensions only, 

 that is, in the case where we have long cylindrical surfaces the 

 properties of conjugate functions can be used for determining the 

 equations of the lines of force and of the equipotential surfaces; 

 see the references in Art. 37 above. However, the purely mathe- 

 matical difficulties of the method are such as to make the analytical 

 calculation of permeances feasible in the simplest cases only. 



In most practical cases, especially in three-dimensional prob- 

 lems, recourse must be had to the graphical method of trial and 

 approximation, in order to obtain the maximum permeance. 

 The field is mapped out into small cells by means of lines of force 

 and equipotential surfaces, drawing them to the best of one's 

 judgment; the total permeance is calculated by properly com- 

 bining the permeances of the cells in series and in parallel. Then 

 the assumed directions are somewhat modified, and the permeance 

 is calculated again, etc., until by successive trials the positions of 

 the lines of force are found with which the permeance becomes a 

 maximum. 



The work of trials is made more systematic by following a pro- 

 cedure suggested by Lord Rayleigh. Imagine infinitely thin sheets 

 of a material of infinite permeability to be interposed at intervals 

 into the field under consideration, in positions approximately 

 coinciding with the equipotential surfaces. If these sheets exactly 

 coincided with some actual equipotential surfaces, the total 

 permeance of the paths would not be changed, there being no 

 tendency for the flux to pass along the equipotential surfaces. In 

 any other position of the infinitely conducting sheets, the total 

 permeance of the field is increased, because through these sheets 

 the flux densities become more uniformly distributed. Moreover, 

 these sheets become new equipotential surfaces of the system, 

 because no m.m.f. is required to establish a flux along a path of 

 infinite permeance. Thus, by drawing in the given field a system 



