An introduction to the mathematical theory of attraction ...

Uniplanar Potential of Ellipse. 161
If X and Y be the components at the point P, whose
coordinates are x and y, of the force due to the ellipse E,
whose uniplanar mass is M, and whose semi-axes are a and b,
by Art. 19, we have, therefore,
2M x 2M y
A= 1=.r3)
X= a' b 2' a a + b'' (13)
where a' and b' are the semi-axes of the confocal ellipse
passing through P.
To find the potential V at an external point, we have
V = C - j (Xdx + Ydy). Since a'2 - b = a2 - b2 = c2 we may
x2 y2
assume a' = c cosh r, b' = c sinh ln, and since -- + - = 1 we
may assume x = a' cos 4, y = b' sin; hence we have
x = c cosh n cos, y = c sinh sin;
and differentiating we obtain
dx = - c cosh 7 sin 4 dÏ + c sinh n cos dn,
dy = c sinh in cos d + c cosh n sin dnr/,
also, we have
x y
a' + ' = cer, = cos y, = sin Ç;
whence, hy substitution, we get
Xdx + Ydy = M { - e-2, (sin 2E d + cos 2 dn) + dn}
= M {2 d (e- 2 cos 24) + d},
and therefore
V= C- -M ({ + e-21 cos 2[}..
At a point at an infinite distance from the centre of the
confocal system n is infinite and e-~ = O, also a' - en, and
y = M log, = M (log 2 - log c - n);
hence we get C = M (log 2 - log c).
M