An introduction to the mathematical theory of attraction ...

154 Surfaces and Curves of the Second Degree.
smallest axis is 2c', to pass through the point and proceed as
in Art. 85. In this manner we obtain
V= 2c~ 2 2 0 2
2C3 J o t \1 + X2( 1 + X+À2u
du (
(1+ ) (1 + À2j2) (9)
If we desire to express the potential at an external point
by means of elliptic functions, we may use equation (8), substituting in that equation 4' for ~i, where c' tan /' = k.
87. Symmetrical Expressions for Potential and
Components of Attraction.-By the transformations
given in Art. 22, we find that at an internal point x, y, z, the
potential V of a homogeneous ellipsoid is given by the
equation
3MW f^_^_ y' ^ \ __^ -_dv
Y=4~ i 4-J ( + b2+v c2+V) /(a2+v-)(b2 +)(c2v)'
(10)
At an external point if 2a', 2b', 2c' be the axes of the
confocal ellipsoid passing through it, we have
3 3M0 X2 3y2 z2 X Z2
4Jo 1 ( 2 " I V2 + v ~,")+ /(2 + V) (b' + ) (c'2 +v)
If we put a'2 + v= a2 + u, we have b + v = b2 + u,
e'2 + = C2 + u, and d =du; also u= o when = o, but
u ='2 a- a2 when v = 0. Hence, if q be the greatest root of
the equation
X2 y2 Z2
+ + + = 1,
a2 +q b2 q c2 q
we obtain
=M }(1 2 y2 + 2 / d)u
+ b2+U c2 + u1 + ) (b2) + 2 + )
(11)