A MEMOIR ON PREPOTENTIALS.
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[607
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point {a,.., c, e), so long as this point in its course does not traverse the material
mula has
surface. For greater clearness we may consider the point as moving along a continuous
curve (one-dimensional locus), which curve, or the part of it under consideration, does
not meet the surface; and the meaning is that the prepotential and each of its
derived functions vary continuously as the point (a, .., c, e) passes continuously along
the curve.
erve that
20. Consider a “region,” that is, a portion of spacfe any point of which can be,
by a continuous curve not meeting the material surface, connected with any other
point of the region. It is a legitimate inference, from what just precedes, that the
prepotential is, for any point (a,.., c, e) whatever within the region, one and the same
function of the coordinates (a,.., c, e), viz. the theorem, rightly understood, is true;.
but the theorem gives rise to a difficulty, and needs explanation.
in com-
Consider, for instance, a closed surface made up of two segments, the attracting
matter being distributed in any manner over the whole surface (as a particular case
s +1 = 3, a uniform spherical shell made up of two hemispheres); then, as regards
the first segment (now taken as the material surface), there is no division into regions,
but the whole of the (s + l)-dimensional space is one region; wherefore the prepotential
of the first segment is one and the same function of the coordinates (a,.., c, e) of the
attracted point for any position whatever of this point. But in like manner the
prepotential of the second segment is one and the same function of the coordinates
(a,.., c, e) for any position whatever of the attracted point. And the prepotential of
the whole surface, being the sum of the prepotentials of the two segments, is
1 for the
consequently one and the same function of the coordinates (a,.., c, e) of the attracted
point for any position whatever of this point; viz. it is the same function for a
point in the region inside the closed surface and for a point in the outside region.
That this is not in general the case we know from the particular case, s +1 = 3, of
o proceed
a uniform spherical shell referred to above.
21. Consider in general an unclosed surface or segment, with matter distributed
over it in any manner; and imagine a closed curve or circuit cutting the segment
once; and let the attracted point (a,.., c, e) move continuously along the circuit. We
may consider the circuit as corresponding to (in ordinary tridimensional space) a plane
curve of equal periphery, the corresponding points on the circuit and the plane curve
negative,
being points at equal distances s along the curves from fixed points on the two
curves respectively; and then treating the plane curve as the base of a cylinder, we
may represent the potential as a length or ordinate, V = y, measured upwards from
the point on the plane curve along the generating line of the cylinder, in such wise
that the upper extremity of the length or ordinate y traces out on the cylinder a
curve, say the prepotential curve, which represents the march of the prepotential.
, . &c. ad
all finite.
The attracted point may, for greater convenience, be represented as a point on the
prepotential curve, viz. by the upper instead of the lower extremity of the length or
ordinate y\ and the ordinate, or height of this point above the base of the cylinder,
ther cases
then represents the value of the prepotential. The before-mentioned continuity-theorem
the pre
attracted
is that the prepotential curve, corresponding to any portion (of the circuit) which
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