348 
PROFESSOR J. H. POYNTING ON THE TRANSFER 
But from the values of P', Q', B/ in (5) we see that 
similarly 
d 2 G d 2 yfc d 2 H 
dz dy dtdz dxdz'dtdy 
_cU<m_clG\ 
clt\dy dz) 
d 2 yjr 
dzdx 
da 
dt 
cU 
dt 
=—=H — (Maxwell, vol. ii., p. 216) 
cl R' dY_db _ d/3 
dx dz dt ^ dt 
dV'_dQ' clc dy 
dz dx dt ^ dt 
Whence the triple integral in (6) becomes 
dt 
clt 
Transposing it to the other side we obtain 
’»!+<» 2 + b “ 
47T 
dt 
clt 
v dz +t J j j (“ it +l3 x) dxd y dz 
dt 
dt 
-f III(X.x+Yy+ Tizyixdydz + j [j (Pp J r Qq-\-'Rr)dxdydz 
= 4 ^jj{W-QV)+w( p V-R /a )+^(Q / «- p/ /3)}dS 
• (7) 
The first two terms of this express the gain per second in electric and magnetic 
energies as in (2). The third term expresses the work done per second by the electro¬ 
magnetic forces, that is, the energy transformed by the motion of the matter in which 
currents exist. The fourth term expresses the energy transformed by the conductor 
into heat, chemical energy, and so on ; for P, Q, It are by definition the components 
of the force acting at a point per unit of positive electricity, so that P 'pdxdijdz or 
P dx/pdydz is the work done per second by the current flowing parallel to the axis of x 
through the element of volume dxdydz. So for the other two components. This is in 
general transformed into other forms of energy, heat due to resistance, thermal effects 
at thermoelectric surfaces, and so on. 
The left side of (7) thus expresses the total gain in energy per second within the 
closed surface, and the equation asserts that this energy comes through the bounding 
surface, each element contributing the amount expressed by the right side. 
This may be put in another form for if (S' be the resultant of P', Q', R', and 0 the 
