OF ENERGY IN THE ELECTROMAGNETIC FIELD. 
347 
P=o y -fe-f-f=c,-fe + F 
ri ' dGr dy\r • ' , r\' 
Q i = az-cx— — — ih =az-cx + Q [ 
E —bx—ay — ^ bx — ay -f P' 
(5) 
where P', Q', P' are put for the parts of P, Q, P which do not contain the velocities. 
Then 
Vu + Qv +P^e = (cy—bz) u + (az—cx) v + (bx — ay) i e+P hi + Q' r +P ’w 
— -{(vc—wb)x-\-(wa—uc)y-\-(ub—va)z} d-Pht-pQ'y-J-Pht; 
= — (Xsc -\-Yy-\- Zz) + Phi+Qh> -{- Phe 
where X, Y, Z are the components of the electromagnetic force per unit of volume 
(Maxwell, vol. ii., p. 227). 
Now substituting in (4) and putting for u, v, w their values in terms of the magnetic 
force (Maxwell, vol. ii., p. 233) and transposing we obtain 
K 
47T 
dP 
dQ 
dR 
dt 
dt 
P ^ +Q rli +Pt M ) dxclydz-\- j j j {(Xx-d h y -f Zz) -j- ( Pp -f Qq + 1 N) 
vaz 
= (P'u-{-($y-\-~R'iv)dxdyclz 
1 
47 tJ J 
4 tt 
, [dy d/3 
dy dz 
, [da. d<y\ rj) , [d/3 da 
f(^-£)+Q'( 5 -^)+e 
dx dy 
dxdydz 
+ 
_ 1 _ 
47rj 
P' ^-P' ~^j dxdydz 
Pit] 
, da d/3 
+^IH(QY-p 
dz 
dxdydz 
Integrating each term by parts] 
(P'/3—Q 'y)dydz-\-~ j (P'y—Phx)c?zdx+^ Jj(Q'a— V'/3)dxdy 
477 
1 
477 
dQ' dV dR' dQ' rfP'l, , 
i J d a--y & zy~ ,x iy +a ih-P s\ dxd y dz 
[The double integral being taken over the surface] 
= T||{i(E^_Q' y )+m( p ' r -R'«)+»(Q'a- p 'j8)}<iS 
1 
477 
a 
dQ' dR' 
dz 
iKV rfF\ , IdY iQ'Xl, , , 
where J, on, n are the direction cosines of the normal to the surface outwards. 
2 y 2 
