THEORY OF ELECTROMAGNETISM. 
729 
way. If (regarding d and k as the independent electric coordinates) we could be 
certain that we had found the full expressions for /, l s , x, x s , both types of external 
electromotive force would he zero. But vve can with considerable certainty say that 
we have not found these completely, so far as they depend upon k and K (electrolysis, 
&c.). On the other hand, it is by no means so obvious that we have not found them 
completely so far as they depend on d and c. Let us then assume that the external 
force (exclusive of frictional forces, of course) of the latter type is zero. If we can 
point to no experimental facts contradicted by this assumption, we may consider that 
the simplification is warranted. 
57. Now (§ 28) the work done per unit volume by the external forces E, e of 
equations (28), (29), § 50 above, while D and d suffer the increments d .D and c/d respec¬ 
tively, is 
SE dD + Se c/d = SE (c/d -j- c/k) + Se c/d 
= S (E + e) c/d + SEc/k. 
Hence, if d and k be taken as the coordinates, the forces of those types would be 
E + e and E respectively. The assumption just made then leads to 
E -f- e = 0, E* + e,, = 0.(1), 
where it must be remembered that the exact meaning of these four symbols is that 
given to them in § 50 above, not the meaning they had previous to that section. If 
we assumed that x was independent of H, equation (l) would be equally true of the 
previous meanings of the symbols. 
We shall now always suppose e to be replaced by — E. With regard to e,. and E„ 
note that by means of equation (l) and equations (30), (31), and (39) of § 50 
OU^u = 0.(2), 
which shows that what we have called the potential is continuous throughout space. 
This will be found to lead to the result that contact-force cannot be explained without 
a slight extension of the independent variables of /, or the assumption that l s is not 
zero. It does not, however, prevent on present assumptions an explanation of the 
Peltier effect. 
IV. Detailed Examination of these Results. 
A. MaxivelVs Results. 
58. The justification of the present theory, where it differs from accepted theory, 
must be based on an examination of its results in detail. First, then, let us compare 
MDCCCXCII.—A. 5 A 
