OF EVERY DEGREE OF TRANSPARENCY. 215 



density. For the average values of these components in the small 

 spaces defined above, we may write 



since it will make no difference whether we take the average before 

 or after the operation of taking the potential. 



6. If we write X, Y, Z for the components of the total electromotive 

 force (electrostatic and electrodynamic), we have 



[X] Ave = - Pot [fl Ave ^-, j_ (3) 



etc., 

 or by (2) 



4?r 2 

 rvi T) /r -pj . r --, 



[X] Ave = 2-Pot[] Ave - 

 etc. 



It will be convenient to represent these relations by a vector 

 notation. If we represent the displacement by U, and the electro- 

 motive force by E, the three equations of (3) will be represented by 

 the single vector equation 



[E] Ave =-Pot[U] Ave -V[ g ] Aye , (5) 



and the three equations of (4) by the single vector equation 



(6) 



where, in accordance with quaternionic usage, V[g] Ave represents the 

 vector which has for components the derivatives of [<?] AT e with respect 

 to rectangular coordinates. The symbol Pot in such a vector equation 

 signifies that the operation which is denoted by this symbol in a 

 scalar equation is to be performed upon each of the components of 

 the vector. 



7. We may here observe that if we are not satisfied with the law 

 adopted for the determination of electrodynamic force we have only 

 to substitute for Pot in these vector equations, and in those which 

 follow, the symbol for the operation, whatever it may be, by which 

 we calculate the electrodynamic force from the acceleration.* For 

 the operation must be of such a character that if the acceleration 

 consist of any number of parts, the force due to the whole acceleration 

 will be the resultant of the forces due to the separate parts. It will 

 evidently make no difference whether we take an average before or 

 after such an operation. 



* The same would not be true of the corresponding scalar equations, (3) and (4). For 

 one component of the force might depend upon all the components of acceleration. 

 Such is in fact the case with the law of electromotive force proposed by Weber. 



