An introduction to the mathematical theory of attraction ...

Ellipsoid. 27
07O
Now Z= p (ri cos 0 + r2 cos 02) sin Oi0 dO ddp,
7r
and Z3 = pj (R1 + R,) cos 01 sin 01 d0i d<c,
Jo Jo
whence Z = Z3, and similarly, X = - X and Y = b Yz.
A purely geometrical method of arriving at these equations will be found in Ex. 12, Art. 24.
If we substitute for X1, Y2, and Z3, their values obtained
from equations (14), we have
3Af P _____ u2 Cd __
X = 3M f1A u2 du
X= x-~x 0(l + 122)U-1 (1+
= _ Jo (1 + X12 u2)2 (1 + X~ 2 tu2)
z- cz -? z (1 + )~ m3)6 (1 + X2U()1'
where A, B, C are constants defined by the equations above.
The expressions in equations (15) for the attraction components of the ellipsoid at a point P whose coordinates
referred to the centre are - x, - y, - z, hold good if P be inside
the external surface of the ellipsoid.
To prove this, supposea similar ellipsoid, whose axes are
2a', 2b', 2c', drawn throf~gh P; then, by Art. 18, the total
attraction at P is the attraction of this ellipsoid. Also, if
M' be its mass,
--- = 3, and X' = X1, X'2= À2;
therefore the coefficients of x, y, z in the expressions for the
attraction components are the same whether P be on the
surface of the ellipsoid or in its interior.
22. Symmetrical Expressions for Components of
Attraction.-Symmetrical expressions for X, Y, and Z may
be obtained in the following manner:
Assume v = c2 tan2O, then
dv = 2c2 sin 0 sec3O dO = 2c2 sin 0 c-3 (c2 + v) dO,