298 PROFESSOR J. H. POYNTING ON ELECTRIC CURRENT AND THE 
The last two terms cancel each other, and we get 
P= 
[[[du 1, , , 1 d[[[(d? dQ dR\l, , , 
“HI J H r dxd y dz -Trr &JJ JU+% + *' 
• (?) 
or if w r e put 
and 
then 
dP dQ dR 
&+*; + &- 4,r ' 5 
Y=[j|- dxdydz. 
p =-4ff C ti dxd y dz -T, 
dx 
( 8 ) 
with similar equations for Q and R, 
If the system is steady ^ are all zero, and then 
dx’ c dy’ 
R= — 
dV 
dz' 
The quantity p, of which V is the potential, will be zero within non-conducting 
homogeneous parts of the field, for there 
KP _KQ KB 
^ 47r 5 ^ 47r ’ ^ 47T ’ 
and 
dP dQ dR = 4^/^ d^\ 
dx dy dz K \dx dy dz) 
since no charges can reside within a homogeneous non-conducting medium. Or, 
stating it in another way, all the induction tubes brought into any part of such a 
medium remain there without dissipation, a charge in a non-homogeneous medium 
being due to unequal amounts of dissipation of induction in different parts of the 
medium. 
But p will have value at surfaces separating dissimilar substances either in the 
insulating or conducting parts of the medium. For in the former the induction is 
continuous, while the intensity is discontinuous, and in the latter the current or rate 
of destruction of induction may be continuous, but the relation between intensity 
and current changes discontinuously with the conductivity. At surfaces separating 
insulators from conductors p may have value, as, for instance, at the surfaces of the 
plates of a condenser with its terminals connected with two points in a circuit, or at 
the surface of an insulated conductor near the circuit. It is also to be noted that 
p will have values at the seat of electromotive force. 
