An introduction to the mathematical theory of attraction ...

160 Surfaces and Curves of the Second Degree.
where > is the area of the ellipse in which the plane meets the ellipsoid. We
may assume
x. y-.. z
sin cos p, - sin O sin, cos
then
Z =P2 {abc2 jJ [cos3 0 sin 0 de d. - q2aibl,
0 a1c2 - q2)
where cos 01 = and aibi = (c - q2)
c C2
whence
rp (c2 - q2)2 pC c;
Z= abC ) _ 2
4 c2 4 7rabc
13. Prove that the attraction of a homogeneous ellipsoid E on a cubical or
spherical portion E1 of its own mass is the same as if the mass of El were concentrated at its own centre of inertia.
If e, v, ' denote the coordinates of any point relative to the centre of inertia
Pi of E1, and xi, yl, zl those of -P relative to the centre of the ellipsoid, and
if L, M, N denote the moments of the attractive forces round the axes meeting
at Pi, we have
L = B fydm-C rlzdm=B Sy (yi+- n ) dmn-CS {q (zî + 5) dm = (B - )SrnS4 dm =.
In like manner M = O, N= 0; hence the proposition is obvious.
88. Uniplanar Potential of Ellipse.-By a method
similar to that employed in Art. 83, a distribution of uniplanar mass can be determined whose potential is zero at all
points in the plane of distribution which are external to an
ellipse, and equal to k 1 - - -- ) at all internal points, the
x2 y2
equation of the ellipse being + - = 1. Since the potential
a b2
is zero at infinity, the total mass is zero (Art. 44), and, as in
Art. 83, we find that:The uniplanar potential of a homogeneous ellipse at any
external point is equal to that of a focaloidal band of equal
uniplanar mass whose boundary coincides with that of the
ellipse.
It is then easy to prove, as in Art. 84, thatConfocal ellipses of equal uniplanar mass have the same
potential at all points in their plane which are external to both.
From this it follows that the attractions of confocal
ellipses of equal mass are equal at all external points.