An introduction to the mathematical theory of attraction ...

Potential of Ellipsoid. 153
Integrating we get
= Vo - 2 (AX2 + By2 + C2), (4)
where Yo is the value of V at the centre of the ellipsoid. To
find the value of Vo we have
Fo= Jpr'drd J Jrj r2 sindO
= p? f2 a2_22 sin 0 dO dp
o
7r 21M a 2Pc2 sin 0 dO d~
2J J a2b2 cos2 0 + c2 sin2 0 (a2 sin2 p + b2 cos2 ',
Treating the integral in a manner similar to that employed
in Art. 21, we get
3M[' du
yr 3 1 du (5)
20 Jo (1 +i)X(22)2 (I + X2 t)2
If, as in At. 24, we put ÀXu = tan ~, we obtain
3-Jr
Vo = M F, (o6
where c = ~, and tan,i = Xi.
Substituting their values for Vo, A, B, C in (4), we
find
3fX 1 2b2 y2 U2 2 2 dx_2c3 Jo + 1 + t 2 - 22 (1 +Ài ( l U2)(1 + à22u2)'
(7)
If the integrals in (7) be expressed by elliptic functions,
we have
V = - F (i)- (Ax + By2 + C), (8)
where A, B, and C, are the coefficients of x, y, and z in equations (27), Art. 24.
To find the potential of a homogeneous ellipsoid at an
external point x, y, z, we suppose a confocal ellipsoid whose