An introduction to the mathematical theory of attraction ...

Conbocal Homoeoids. 163
coordinates x, y, z, and x', y', z', referred to the axes of the
quadrics, satisfy the equations
x o' y y' z z'
a a" b" c c'
Two coaxal ellipsoidal shells Z and 0' are made up of corresponding points if the codirectional axes of the boundaries of
5 are proportional to those of ('.
To prove this, let a, b, c, be the semi-axes of one boundary
of 5, and Xa, yib, vc, those of the other, then the semi-axes
of the boundaries of 0' are a', b', c', and a', ub', vc', and if
we assume
x x' y y z z'
a a" b =b"' c c
when the point x, y, z is on a boundary of 5, the point x', y', z
is on the corresponding boundary of.'.
If any two points, P1 and P,, be taken on the ellipsoid
whose semi-axes are a, b, c, and the
corresponding points P', and P',,
2 on the confocalellipsoid whose semi-./ ~ ) \ aaxes are a', b', c', the distances P, P',
and P'1 P2 are equal.
For let a, yj, zl, be the coordinates of P,, with a similar notation
for the other points, then,
\ P xPi22 = X2 + y2 + X12 + 22 +y'22 + z 22
- 2 (xx 2 + yLy2 + ) i
but
2 + y 2 + 2 _a æ/2 b 2 2,12
a2_ 2 22 2 1
a2 a2 + x2 + yb2 + b" c + Cz
b" 2 + '
a bb CI
'2 2
= a 6{ + Y, ' 'y + ~z.
and in like manner
X 2 + y/22 + 22 = az 2 -a2 +X+ 22 + y + 22,
M2