An introduction to the mathematical theory of attraction ...

Examples. 35
The components of the attraction o? an ellipsoid at a point x, y, z on its
surface are of the form - Ax, - By, and - Cz. If we substitute these expressions for X, Y, Zin (a), Ex. 5, we have
(A - 2) dx + (B- w2)ydy+ zdz = O.
The differential equation of the surface of the ellipsoid is
xdx dy
x+ -. zdz = O,
1 + 12 1+ h22
and this equation is identical with the former provided that
c= (1 + A12) (A - w2) = (1 + À22) (B - W2)
If we eliminate w2 from these equations, we have
(1 + Ai2) (1 + À22) (A - B) = (À22 - 12) C.
Substituting for A, B, C from (15), and putting
/(1 + À12 2) (1 + A22 22) = U,
we get
1 U)2du 2 du
(1i+ ) (1 + X2) + 1 2 2 dA1
Transposing and reducing, we have
(xii - xi5 i (i _ U) (1 - 12 A22 t2) U2 dO.
If Ai be given, one solution of this equation is A2 = A1, which corresponds to
an ellipsoid of revolution. If this solution be rejected, and if we put
(A + x) 1 (1 - u2) (1 À- 2 Ai2 Z2)) n2 d (
(i + 2) - U3 =f(À2),
we see that f(À2) is positive when À2 = 0, and negative when = O. Hence
f (2) must vanish for some positive value of A2. This value of À2 gives a real
ellipsoid satisfying the conditions of the question, provided the corresponding
values of xi and w2 are real and positive. From the equations of condition given
above we find
2 = A _ _ 12 3M f1 (1 - 2) U2 du
+ h1i2 1 + A,2 c3 Jo (1+Ai2 u)U
whence it appears that if xA be assigned, the corresponding value of w2 is real and
positive. The above theorem is due to Jacobi.
7. If the figure of a revolving mass of homogeneous fluid in relative equilibrium be an ellipsoid, its shortest axis must be the axis of rotation.
D 2