THEOKY OE ELECTROMAGNETISM. 
745 
demand is made by the ordinary theory. IT and A' on the one hand, must be deter¬ 
mined in terms of C' and I', on the other, by means of particular relations. 
Let us suppose—merely to get rid of the dashes—the standard position to coincide 
with the actual position. One difference between Maxwell’s theory and the ordinary 
theory is that according to the latter it is assumed that each individual magnetic 
molecule and each elementary current has its own influence—independently of the 
rest—in producing terms in A and H. Thus, H consists of two parts, the flrst 
depending only on the magnetism and the second only on the currents. The 
firstT= — Vfl, where O = — JjjSlVwhs [Maxwell’s ‘Elect, and Mag.,’ 2nd ed., 
§ 383, equation (3)], where u~ l is the distance of the element e/? from the point 
under consideration, and where in the differentiations of Vu the end of u~ i at the 
element d s is supposed varied. The second part is obtained on the assumption that 
each closed current causes a term in H which the corresponding magnetic shell would 
cause. The second part is thus found to be Vjjj? tCcta"" The first part of A is 
supposed to depend on I in the same way as the A, called the vector potential, of 
Part III. of Maxwell’s treatise depends, i.e., = jjJVTVuJs [§ 405, equation (22)]. 
The second part, as with H is obtained by assuming that any closed current will 
cause a term in A equal to the term in A that would be caused by the corresponding 
magnetic shell. The second part is thus found to be jjjwChs.t We will suppose 
that the ordinary theory also admits that A is arbitrary in containing a term Viv, 
where w is a scalar. (This is only to render the comparison with the present theory 
simpler. Perhaps it ought to be said that the A thus obtained in terms of I and C 
on the ordinary theory is found to satisfy the conditions SVA = 0, [SchSA],, + h — 0, 
and that the present theory only agrees with the ordinary theory if we arbitrarily 
impose those relations.) All this may be expressed thus. Defining A 0 and H by 
A 0 = fjjwCefe, n = - ([jsiVcoh.(33), 
we shall have 
A = A 0 +||| YI Vitrfs + Vw .(34) 
H = — vn + VA 0 .(35). 
If q be any quaternion function of the position of a point which may be dis¬ 
continuous at certain surfaces, we have 
* The magnetic force at an external point due to a shell of strength c = cYjJSdXVn = — cJJ Sd2V.Y?t 
= cfjVdSV.Vtt [since V 2 m = 0] = cj+V?t = cV|W/>. The reason for the change of sign in V on crossing 
the integral sign is that when outside one end and when inside the other end of u~ l is naturally 
supposed in the differentiations of Vu to vary. 
- f- This is not inconsistent with §§ 616. 617 of ‘ Elect, and Mag.,’ 2nd edit., for there Maxwell is 
considering the two parts together. 
MDCCCXCII.—A. 5 C 
