243] EQUATIONS OF ELECTROMAGNETIC FIELD. 505 



From the equality of the line-integrals we must not conclude 

 the equality of the integrands, for the line-integral of any lamellar 

 vector point-function around a closed path vanishes. We accord- 

 ingly obtain 



F =- a ! +r > 



where (Z', F', Z') is a lamellar vector. If X, F, Z denote the 

 whole electric force, when the state of the magnetic field is not 

 changing it becomes the electrostatic force, so that the components 

 X', Y', Z' must be the negative derivatives of the electric potential. 

 Accordingly the equations are 





_ 



dt a* 



These are the equations as given by Maxwell*. We shall 

 however prefer the form (5), not containing either potential, as 

 introduced by Heavisidef and Hertz J. Since the electrostatic 

 field has no curl, it need not be considered separately in equa- 

 tions (5). 



If however there are impressed electromotive forces X, Y', Z' 

 not of electrostatic origin, such as those due to chemical or 

 thermal effects, and X, F, Z still denote the total field, we must 

 replace X, Y, Z in equations (5) by X - X' ', F- F', Z-Z'. 

 (Heaviside, Vol. i. p. 449.) 



In a closed conductor undergoing electromagnetic induction 

 there are not necessarily differences of electric potential, for 



* Treatise, Art. 598, equations (B). 



t "Electromagnetic Induction and its Propagation." Electrician, Feb. 1885, 

 Papers, Vol. I., p. 447, eq. (20). 



J " Die Krafte elektrischer Schwingungen behandelt nach der Maxwell'schen 

 Theorie." Wied. Ann., 36, p. 1, 1889. Jones's trans., p. 138. 



