An introduction to the mathematical theory of attraction ...

Ellipsoid. 25
that Z3 denotes the total resultant attraction at the extremity
of the semi-axis c.
We obtain, in like manner,
AI= p =47a ~c -- — " - — Ud (9)
Jo = 4a2 b + (2 - bc) 2} 2 { + (2 - a2) })' ()
where X, and Yz denote the attractions at the extremities of
the semi-axes a and b. The expression for Z3 in equation
(8) can be put into a more convenient form by taking the
factor c2 in the denominator outside the integral sign, and
by putting
a2 2
c2_ 2 b2 02
we have, then,
47rpab 12d (11)du
c Jo ( + Xi2u2) (1 + X22t 2) 1)
To make X, depend on ÀA and X2, assume
a 2 C 3
a2 - -2, then d=- -, A2 5
/ \2 a2
and _ =
2 (1 + A12 V2)
Substituting for u in terms of v in equation (9), since the
limiting values of v are the same as those of u, we get
47rpab a {1 _ v2 dv(
XI c J V 1J
c cJ ( + Xlv2)2 ( + X2V2)'
If a and b be interchanged, X, and AX become Y' and X,;
hence we obtain
4rrpab b v1 dv (13)
L2 1 3 (1 2)) +
C C X 12 V' ~,292)2(1 + à22~18)