PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 279 
is —47r, when (|, >j, Q is included in the limits of integration ; and therefore the value 
of the triple integral, in the expression for V, is — 4 t(^). Hence, according as the 
point (I, Q is external or internal with reference to the magnet, the potential at it 
is given by the expressions 
d^ 
1 
('•) v=[^. 
or (2.)» + 
dy 
1 
_A 
dy 
d 
-l-47r(®) 
(VII. 
These agree with the expressions obtained above in § 76 ; the same double integral 
with reference to the surface being here expressed symmetrically by means of 
rectangular coordinates. 
78. The value of p at any point in the surface of the magnet, which, as appears from 
the preceding investigations, is all that is necessary for determining the potential due 
to a lamellar magnet at any point not contained in the magnetized substance, may, 
according to well-known principles, be determined by integration, if the tangential 
component of the magnetization at every point of the magnet infinitely near its sur- 
face be given. It appears therefore that, if it be known that a magnet is lamellarly 
magnetized throughout its interior, it is sufficient to have given the tangential com- 
ponent of its magnetization at every point infinitely near the surface or to have 
enough of data for determining it, without any further specification regarding the 
interior distribution than that it is lamellar, to enable us to determine completely 
its external magnetic action. This conclusion is analogous to a conclusion which 
may be drawn, for the case of a solenoidal distribution, from the expression obtained 
in § 51, for the potential of a magnet of any kind. For, from this expression, we have, 
according to § 74, the following in the case of a solenoidal distribution : — 
(VIII.); 
from which we conclude, that without farther data regarding the interior distribution 
than that it is solenoidal, it is sufficient to have given the normal component of the 
magnetization at every point infinitely near the surface to enable us to determine the 
external magnetic action. Yet, although analogous conclusions are thus drawn from 
these two formulee, the formulae themselves are not analogous, as the former (that of 
§ 51) is applicable to all distributions, whether solenoidal or not, and shows precisely 
how the resultant magnetic action will in general depend on the interior distribution 
besides the normal magnetization near the surface, according to the deviation from 
* It may be proved that the force derived from a potential having the same expression (VII.) (1.) as for ex- 
ternal points, is, for any internal point, the force at a point within an infinitely small crevass perpendicular to 
the lines of magnetization ; as it is easily shown that the differential coefficients of 4'jr(<p) are the rectangular 
components of the force at such a point due to the free contrary polarities on the two sides of the crevass. 
