104 Systems of Conductors [OH. iv 



and since this equation is true for all displacements and therefore for all 

 values of Sft, Sft, . .., it follows that each coefficient must vanish separately. 



Thus | = 0, or 



aft aft 



3TF 



As we have seen, -- 1 - is the mechanical force tending to increase ft, 

 0fi 



and this has now been shewn to be equal to -^- , which is expression (49) 



ui 



with the sign reversed. Thus the mechanical force, whether the charges or 

 the potentials are kept constant, is 



P Vf + 2 |f F.F. + ...) ..................... (52), 



i Ogi / 



a form which is convenient when we know the potentials, but not the 

 charges, of the system. 



In making a small displacement of the system such that ft is changed 



rl T/F 



into ft + dft, the mechanical work done is -^ dft. If the potentials are kept 



0i 



constant the increase in electrical energy is -^r^ft. The difference of these 





 expressions, namely 



V aft aft 



represents energy supplied by the batteries. From equation (51), it appears 



that this expression is equal to 2 ^ & F <^ft> so that the batteries supply energy 



tffi 



equal to twice the increase in the electrical energy of the system, and of this 

 energy half goes to an increase of the final electrical energy, while half is 

 expended as mechanical work in the motion of the conductors. 



Introduction of a new conductor into the field. 



121. When a new conductor is introduced into the field, the coefficients 

 >n> ji2, -, #11, #12 > are naturally altered. 



Let us suppose the new conductor introduced in infinitesimal pieces, 

 which are brought into the field uncharged and placed in position so that 

 they are in every way in their final places except that electric communication 

 is not established between the different pieces. So far no work has been 

 done and the electrical energy of the field remains unaltered. 



Now let electric communication be established between the different 

 pieces, so that the whole structure becomes a single conductor. The separate 



