504 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



convert it into a surface-integral which, in virtue of (i), must be 

 equal to the negative time-derivative of the surface-integral in (2). 



/ \ ((\ft z aF \ / \ ft x ^ Z \ / x fiY dX\\ . 

 (4) [His -- -5- cos (nan + -= -- ~- cos (nv) + [-= --- ^- }\ dS 

 JJ(\dy d*/ \dz dJ \dx dyj) 



S ^^ d8 ' 



As we assume that the circuit does not change geometrically 

 with the time the differentiation with respect to t may be passed 

 under the sign of integration, and operates only on the quantities 

 8, SR, 91. Since the two surface-integrals may be taken over the 

 same surface, and the equality holds for any portion of surface 

 whatever (as we may choose any cap over any circuit), the 

 integrands are necessarily equal at all points of space, necessi- 

 tating the equations 



_as _a^_a_F 



dt ~8y a*' 



_89tt_az_az 



" dt ~ 8* das' 



_ = _ 



dt ~~ dx dy ' 

 These equations, which are more compactly expressed by 



(6) _ 



are the general equations of induction, and are justified because of 

 their leading, by the reverse process, to the equation (i), which is 

 directly verified by experiment. A direct experimental verification 

 of equations (5) has been given but recently. 



If we wish to introduce the vector-potential belonging to the 

 magnetic induction, by 226 we have the alternative expression 

 forp* 



(7) p = l(Fdx + Gdy + Hdz). 



Comparing this now with the line-integral in (3) gives us 



(8) l(Xdx + Ydy + Zdz) = - ^ f(Fdx + Gdy + Hdz). 



* See the definition of p following equation (3), p. 469. 



