Maxwell. 273 



through that surface." The electromotive force of induction at 

 the place (x, y, z) is - d&/dt : as Maxwell said, " the electromotive 

 force on any element of a conductor is measured by the 

 instantaneous rate of change of the electrotonic intensity on 

 that element." From this it is evident that a is no other than 

 the vector-potential which had been employed by Neumann, 

 Weber, and Kirchhoff, in the calculation of induced currents ; 

 and we may take* for the electrotonic intensity due to a 

 current i r flowing in a circuit s' the value which results from 

 Neumann's theory, namely, 



., f *s' 

 = t' 



} r 



It may, however, be remarked that the equation 



curl a = B, 



taken alone, is insufficient to determine a uniquely ; for we can 

 choose a so as to satisfy this, and also to satisfy the equation 



div a = ;//, 



where i// denotes any arbitrary scalar. There are, therefore, an 

 infinite number of possible functions a. With the particular 

 value of a which has been adopted, we have 



3 ., f dx' 8 f dy' 8 ., f dz 

 div a = - i \ - + ^' -2- + - i' \ 

 te I' r fy ) 8 , r dz J, r 



., 

 * 



= 0; 

 so the vector-potential a which we have chosen is circuital. 



In this memoir the physical importance of the operators 

 curl and div first became evidentf ; for, in addition to those 

 applications which have been mentioned, Maxwell showed that 



* Cf . p. 224. 



t These operators had, however, occurred frequently in the writings of Stokes 

 especially in his memoir of 1849 on the Dynamical Theory of Diffraction. 



T 



