Whenever, therefore,
is discontinuous at surfaces, the expressions
and
must be regarded as implicitly including the surface-integrals
and
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respectively, relating to such surfaces, and the expressions
and
as including the surface-integrals
and
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respectively, relating to such surfaces.
101. We have already seen that if
is the curl of any vector function of position,
(No. 68.) The converse is evidently true, whenever the equation
holds throughout all space, and
has in general a definite potential; for then
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Again, if
within any aperiphractic space
contained within finite boundaries, we may suppose that space to be enclosed by a shell
having its inner surface coincident with the surface of
We may imagine a function of position
such that
in
outside of the shell
and the integral
for
has the least value consistent with the conditions that the normal component of
at the outer surface is zero, and at the inner surface is equal to that of
and that in the shell
(compare No. 90). Then
throughout all space, and the potential of
will have in general a definite value. Hence,
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and
will have the same value within the space
[1]102. Def.—If
is a vector function of position in space, the Maxwellian[2] of
is a scalar function of position, defined by the equation
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(Compare No. 92.) From this definition the following properties are easily derived. It is supposed that the functions
and
are such that their potentials have in general definite values.
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- ↑ [The foregoing portion of this paper was printed in 1881, the rest in 1884.]
- ↑ The frequent occurrence of the integral in Maxwell's Treatise on Electricity and Magnetism has suggested this name.