270 ELECTROSTATICS. [FT. II. CH VI. 



Then the work done in a displacement S^, ... S<j> m is 



(i) ^Sfc + <,$& ...... +<E>*m, 



and if no energy is furnished to the system this work must be done 

 at the expense of the electrical energy W and 



(2) 



In the differential STF we may use according to circumstances 

 either of the three forms 



TT.r = iS..F w W e or W v , 



which are of course identical, though expressed in terms of different 

 variables. If we choose W ev the total differential 



(3) 



does not contain the S<'s explicitly. For neither the coefficients p 

 nor q appear in W eV . However, the Se's and SF's are not inde- 

 pendent, being connected by either set of equivalent linear relations 

 (4) or (10) above, which in the coefficients q or p involve the para- 

 meters <, consequently we may eliminate either the SF's or Se's, 

 and replace them by 8<f>'a. 



Now we see by the relations 



dW v 



= e 



that the functions W e and W v are reciprocal functions ( 63) 

 with respect to either set of independent variables e l} ...e n , or 

 Fj, ... V n , containing also the independent variables <f>, corre- 

 sponding to the variables z of 63. Accordingly by the last of 

 equations (5), 63, 



(4) e ~ ~ 



If the conductors are insulated, so that all the charges are 

 constant, we use the form W e , so that any force 4> s has the value, 

 from (2), 



(t\ ct> = 



d<t>s ' 



The system tends to move so as to diminish the energy. 



If on the other hand the potentials are maintained constant we 

 must use the form W v . In this case we must supply energy from 



