Vol.. 8, 1922 
PHYSICS: H. A. LORENTZ 
335 
an instant t at which the final state has been practically reached. The 
final values of D, C, etc. are denoted by D, C, etc. 
The main point in the following reasoning is that, when the interval 
is infinitely short, the values of D and B at the instant / = 0 are like- 
wise infinitely small, so that we may reckon with D = 0, B = 0 for ^ = 0; 
moreover, at ^ = 0 the current C cannot be infinite and we may neglect 
the work of the forces Fi and F2 during the interval A^. In the time- 
integral representing the work (up to the time t), we may take t = 0 for 
the lower limit. 
Pfoc/.— During the interval At the finite forces Fi and F2 cannot produce 
an infinite E. Therefore, according to (5) B can at the best have a finite 
value. But then (because at the beginning of A^, B = 0) B must be 
infinitely small until t = 0. The same is true of H (because connected 
with B by (3)). Therefore, on account of (4), D + C must be infinitely 
small. But, as (2) shows, C cannot be infinite; therefore D also cannot 
be infinite, and D must be infinitely small at t = 0.^ We can conclude 
now that both the integrals J^(Fi.t>)dt and J^{F2,C)dt which repre- 
sent the work of the electromotive forces, when taken over the infinitely 
small time A^, have infinitely small values ; indeed, neither (Fi . D) nor 
(F2.C) becomes infinite in the interval. 
dS is the element of volume. Integrations with respect to dS are ex- 
tended over the whole field, when necessary to infinite distance. 
r 
The electric energy in the final state is U = ^/^ J — d S, or, on ac- 
count of (1)^ 
U= i J({E+Fi}.D)J5 (8) 
The magnetic energy in the final state is T = 3^ (H. B)<i5. 
The loule heat per unit of time in the final state is 
for which we may also write^ J^(F2.C)dS. 
If the Joule generation of heat started suddenly at ^ = 0 at the rate 
given by the last expression, the amount of heat developed until the time 
/ would be^ 
W = £ dtJi¥2.C)dS (9) 
The work of the impressed forces up to the time t is given by 
^ = £dt J (Fi. i>)dS -F jT' dt J(F2. C)dS, (10) 
