464 Induction of Currents in Continuous Media [CH. xv 



have the same value whatever path is chosen. Thus ^ as defined by equa- 

 tion (469) is a single-valued function of the coordinates of P. Let us now 

 move from P a distance dx parallel to the axis of x. The increase in the left- 

 hand member of equation (4G9) is 



dF 



while that in the right-hand member is 



&& , 



^ doc. 



dx 



These two expressions must be equal, hence we must have 



*--- ........................... <> 



v dG 3^ /m 



while similarly 1= -- = -- ^ ..... ...................... (471), 



dt dy 



~- ,, ~ (472). 



dt dz 



The function M* has, so far, had no physical meaning assigned to it. 

 Equations (464), (465), (466) shew that the electric force (Z, F, Z) can be 

 regarded as compounded of two forces : 



/ dF dG dH\ . . 



(i) a force -3- , -j- , -= arising from the changes in the mag- 



\ dt af ^*y : dt dt j 



netic field; 



(ii) a force of components f ^ , ^ , -^ J which is present when 

 there are no magnetic changes occurring. 



We now see that the second force is the force arising from the ordinary 

 electrostatic field. Thus ty differs by a constant only from the electrostatic 

 potential. W T e shall now suppose the point of equation (463) to be a point 

 at infinity. Then ^ = at infinity, so that we may now identify M/ 1 with the 

 electrostatic potential. 



531. If we differentiate equation (472) with respect to y, and (471) with 

 respect to z, and subtract, we obtain 



___ = ___ 



cy "bz dt \ dy 



da 



~dt' 



by equation (467), bringing us back to the system of equations (464) (466). 



