and, if the space is periphractic, that the surface-integral of
vanishes for each of the bounding surfaces.
The existence of the minimum requires that
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while
is subject to the conditions that
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and that the tangential component of
in the bounding surface vanishes. In virtue of these conditions we may set
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where
is an arbitrary infinitesimal scalar function of position, subject only to the condition that it is constant in each of the bounding surfaces. (See No. 67.) By substitution of this value we obtain
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or integrating by parts (No. 76)
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Since
is arbitrary in the volume-integral, we have throughout the whole space
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and since
has an arbitrary constant value in each of the bounding surfaces (if the boundary of the space consists of separate parts), we have for each such part
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Potentials, Newtonians, Laplacians.
91. Def.—If
is the scalar quantity of something situated at a certain point
the potential of
for any point
is a scalar function of
defined by the equation
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and the Newtonian of
for any point
is a vector function of
defined by the equation
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Again, if
is the vector representing the quantity and direction of something situated at the point
the potential and the Laplacian of
for any point
are vector functions of
defined by the equations
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