257] EQUATIONS OF ELECTROMAGNETIC FIELD. 555 



cause at the rate g) r-, and in the same manner from its ^-projection 

 at the rate 3 ^- But the area of the X-projection of the element 



O /O O 



is also increasing, at the rate ^- in the F-direction and ~ in the 

 ^-direction. From this cause the flux increases at the rate 



We have therefore to replace the term ^- in equations (A) by 



ut 



the sum 



, O , 



(2) ^r + a^- + 5- +7 + ^- ^- 



dt dx dy ' dz (dy dz) ( dy dz 



We have thus added in virtue of the motion two parts, the 

 first of which is the component of the curl of the vector whose 

 components are 



79-03, 3--y /3-g), 



that is the vector product of the induction of the field and the 

 velocity. The last term is the component of the velocity times 

 the divergence of the induction. We may therefore abbreviate 

 our equations which replace both (A) and (B) thus 



(A") A |?f + curl Vgu + v div g + 4w?l = curl H, 



(ot ) 



<B") - A + curl VS3u + v div 33 = cur* F. 



(dt } 



We may interpret the meaning of the new terms physically 

 thus. In the equation (B /x ) the term 



produces a part of curl.F. Two vectors having the same curl 

 differ only by a lamellar vector (| 223). Consequently the motion 

 gives rise to an electromotive force AV$$v = J.Vu53 perpen- 

 dicular to the magnetic field and to the direction of the motion. 

 This is the ordinary electromotive force of induction, and its mag- 

 nitude may be specified as equal in any element of conductor 



