250 THE MAGNETIC CIRCUIT [ART. 71 



the conservation of energy to this displacement. From the 

 equation so obtained the component of the force in the direction 

 of the displacement can be calculated. Taking other displace- 

 ments in different directions, a sufficient number of the com- 

 ponents of the forces are determined to enable one to calculate 

 the forces themselves. Since the forces in a given position of 

 the system are perfectly definite, the result is the same no matter 

 what displacements are assumed, provided that these displace- 

 ments are possible, that is, consistent with the given conditions 

 of the problem. Therefore displacements are selected which 

 give the simplest formulae for the energies involved. We have 

 had two applications of this principle in the preceding article, 

 in deriving the expressions for the tension and the compression 

 in the field, by giving the simple magnetic circuit the proper 

 " virtual " displacements. In applying this method, not only 

 the mechanical displacement has to be specified, but also the 

 electric and the magnetic conditions of the circuit, in order to 

 make the energy relations entirely definite. Thus, in the pre- 

 ceding article, the electromagnetic condition was <D = const. 1 



First let us take the case when the partial linkages are neg- 

 ligible; then according to the third eq. (99), the stored energy is 



....... (176) 



where (R is the reluctance of the circuit. Let F be the unknown 

 mechanical force between two parts of the magnetic circuit at 

 a distance s, and let one part of the system be given an infini- 

 tesimal displacement ds. Let F be considered positive in the 

 direction in which the displacement ds is positive. The mechan- 

 ical work done is then equal to Fds. As in the preceding article, 

 let this displacement take place with a constant flux, so that 

 there is no interchange of energy between the magnetic circuit 

 under consideration and the electric circuit by which it is excited. 

 Then the work is done entirely at the expense of the stored energy 

 of the magnetic circuit, and we have : 



Fds=dW m =-dW 8 , ..... (177) 

 where dW m is the mechanical work done. The sign minus before 



1 The principle of virtual displacements is much used nowadays in the 

 theory of elasticity and in the calculation of the mechanical stresses in the 

 so-called statically-indeterminate engineering structures. 



