THEORY OF ELECTROMAGNETISM. 
739 
course followed here is, of course, to abide by what the present theory leads to, and 
then to choose that particular interpretation of the above passages which appears 
least at variance with our results. [It will be seen from the above that the state¬ 
ments at the end of § 50, above, are true, viz., that the potential v of the present 
theory is certainly not [equation (22), § 62] the same as Maxwell’s potential z, and 
that without some such assumption as SV'A' = 0, [ScZX'A']^ = 0 our potential, 
like Maxwell’s, is indefinite apart from an arbitrary additive constant. This 
question of the arbitrariness of the potential is one merely of terms.] 
It is, perhaps, unnecessary now to say that the position I wish to maintain is that 
Maxwell has not investigated in a perfectly general manner the consequences of his 
own theory, and that, consequently, some of his general equations may prove incon¬ 
sistent with that theory. Equation (12) I hold to be such an equation. So little 
right, indeed, has he to put this down as one of his general results that it is, I hold, 
inconsistent with other parts of his treatise. For instance, if the equation were 
consistent with equation (4), § 640 (‘ Elect, and Mag.,’ 2nd edition), we should 
have V/SH/I' = V'OSVT, which is certainly* not the case in general on Maxwell’s 
theory. I shall not, then, compare the mechanical results of the present theory with 
equation (12) at all, but shall adopt the simpler process of comparing the stress which 
results from the present theory with that which Maxwell obtains in Chapter V. of 
Part I., and Chapter XI. of Part IV. 
67. Before this comparison another matter must be considered. Maxwell, in 
accordance with, I think, universal custom, supposes that a molecular couple exists 
due to magnetism. In the first place this extraordinary exception to our ordinary 
* As might be expected, the relation is true in very many important problems whose details have been 
worked out, but it is not true in general, even when there are no currents. Dropping the special nota¬ 
tion of this paper for the moment, let r, x have their usual Cartesian meanings. Denote differentiations 
with regard to r by dashes. Let E be any function of r. If there be no currents, and if 
Q = xF, 
then will 
— H = VQ = iE + pxF'/r, 
and [from the relation SV (H + 4 ttI) = 0, which is the only equation to be satisfied] 
I = i (VF' + 3E)/4v. 
In this case 
IvVjSIHj = - 4vSIV.Vfi = OF' + 3F) {i2xF'/r + P [F + x*d (F'/fificfr]/?-}, 
and 
4/tVOSVI = — O/r) d OF' + 3F)/Jr,fiF -f pxF'/r}, 
which are clearly not in general equal. The above expression for I is, of course, not the general one for 
this case, as we may add to it a term Wo- where a is any vector. Also, it is assumed that F is such 
that both H and I are everywhere continuous, i.e., F and F' are everywhere continuous. For instance, 
put F = 0 — a )~ from r = 0 to r = a, and F = 0 from r = a to r = oo. 
5 B 2 
