757]
261
757.
ON A SMITH’S PRIZE QUESTION, RELATING TO POTENTIALS.
[From the Messenger of Mathematics, vol. xi. (1882), pp. 15—18.]
A spherical shell is divided by a plane into two segments A and B, one of them
so small that it may be regarded as a plane disk: trace the curves which exhibit the
potentials of the two segments and of the whole shell respectively, in regard to a point
P moving along the axis of symmetry of the two segments.
Criticise the following argument:
The potential of the segment A in regard to a point P, coordinates (x, y, z), is
one and the same function of (x, y, z) whatever be the position of P; similarly the
potential of the segment B in regard to the same point P is one and the same function
of (x, y, z) whatever be the position of P: hence the potential of the whole shell in
regard to the point P is one and the same function of (x, y, z) whatever be the
position of P.
The question is taken from my memoir “ On Prepotentials,” Phil. Trans, vol. 165
(1875), pp. 675—774, [607]; and the figure of the curves is given p. 689*. There is
no difficulty in tracing them by means of the expression for the potential of a plane
circular disk in regard to a point on its axis of symmetry: it was in order that
they might be so traced, that one of the segments was taken to be small; but I
had overlooked the circumstance that the formula for the disk is in fact only a
particular case of a similar and equally simple formula for the spherical segment:
viz. (as was found in one of the papers) the potential of a spherical segment in
regard to a point on the axis is = ~~~~ (pi ~ P2), where p, p x , p 2 are the distances of
the attracted point from the centre of the sphere and from the centre and the circum
ference respectively of the segment. The segments might therefore just as well have
been any two segments whatever, or (to take the most symmetrical case) they might
have been hemispheres.
As to the argument: the assertion in regard to the potential of the segment
[* This Collection, vol. ix. p. 333.]
