THEORY OF ELECTROMAGNETISM. 
717 
the present we assume, we are considering that part of L contributed by a finite 
portion of matter we must retain the part of the surface integral due to the true 
boundary of the portion (jj$ of § 42). Thus the part of SL due to (6) is 
- ||[ SA SC (U - (4 v)~ 1 1| SA SH dt 
= - | HI SA SD ds + HI SA SD ds - (4tt)- 1 |J SA SH dt. 
Collecting terms we have for any finite portion of matter 
SL = - IfffjDjy-Sp' + S8D( C V« + A)} ck 
+ fj]{Sy[D„(p- + V'W) - mfV,] - SSd„v; 
+ S 3d ( d c Vl/dt + A - D VZ)} ds 
S Sp'f dt - (47T)- 1 1| SA SH dt .. (9). 
B. The Free Energy and Rate of Increase of Intrinsic Energy for any Finite portion 
of Matter. 
46. We now see from the principle enunciated in § 33, above, that the modified 
kinetic energy for all space is given by 
2 L = - iff + SC ( c Vl + A)} ck. 
Now 
47r||fsCAds = JIJsAVHcZs [§ 25, eq. (13)] 
= J|JSHVAds +f|SHAc& [§ 5, eq. (4)] 
= [§ 26, eq. (19)], 
the surface integral vanishing by § 25, eq. (15) and § 26, eq. (20). Thus 
2$. = - |||{D w p' 3 + SC c V/+SBH/47t} ds = ffj(Z + X) ds . . (10), 
where X is in value, but not in form (since we suppose it expressed in terms of the 
