in Media of every degree of Transparency. 111 
since it will make no difference whether we take the average 
before or after the operation of taking the potential. 
f we write X, Y, Z for the components of the total elec- 
tromotive force (electrostatic and electrodynamic), we have 
“ A dave 
[X]ave = — Pot [&Jave — a (3) 
ete. 5 
or by (2) 
47? v 
[X]ave = <F- Pot [are — Lass, (4) 
ete. 
It will be convenient to represent these relations by a vector 
hotation. If we represent the displacement by YJ, and the 
electromotive force by E, the three equations of (3) will be 
Tepresented by the single vector equation 
[Elave = — Pot [Ulave ae i [Waves (5) 
and the three equations of (4) by the single vector equation 
4 2 
[Elave = Fy Pot [Ulave — 71d]ave (6) 
where, in accordance with quaternionic usage, 7[q] Ave Tepre- 
Sents the vector which has for components the derivatives o 
[@]ave with respect to rectangular codrdinates. The symbol 
Pot in such a vector equation signifies that the operation which 
18 denoted by this symbol in a scalar equation is to pe 
ormed upon each of the components of the vector. 
‘. We may here observe that if we are not satisfied with the 
aM adopted for the determination of electrodynamic force, we 
ant 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. 
8. Let us now examine the relation which subsists between 
the values of [E]ave and [U]ave for the same point, that Is, 
between the average electromotive force and the average dis 
Placement in a small sphere with its center at the point consid- 
F *The same would not be true of the corresponding scalar equations, @) real 
a = component of the force might depend se . Deane cnaek: 
pon uch is in fact the case with the law of electromotiv 
