189-192] 
Ellipsoidal Configurations 
209 
where 6 is given by equation (190’3). Equating coefficients of x 2 , y 2 and z 2 , 
gives the value of 6 . Thus the three equations just written down contain 
shall be a figure of equilibrium under the rotation on. 
191. Subtracting corresponding sides of the first two equations, we obtain 
Thus the necessary equations can be satisfied in two ways, either by taking 
The first condition requires that the body shall be spheroidal, the cross 
sections by the axis of rotation ( Oz ) being circles. If, however, the second 
condition is satisfied, a and b are not in general equal (as we shall shortly 
see), and the configurations are ellipsoids with three unequal axes. 
Clearly the two equations (191T) and (19T2) represent two linear series 
of equilibrium configurations, which are spheroidal and ellipsoidal respec 
tively. The configurations which form the first series are commonly known as 
Maclaurin’s spheroids; those which form the second as Jacobi’s ellipsoids, 
their existence having first been demonstrated by Jacobi* in 1834. We shall 
examine these two linear series in turn. 
Maclauriris Spheroids. 
192. When a = b equations (190-4) and (190"5) become identical, and the 
three equations (190"4) to (1906) reduce to two: 
this equation is seen to be equivalent to the three separate equations 
6 
(190-4), 
2 -rr<ypabc b‘ 
(190-5), 
(1906). 
By addition of corresponding sides we again obtain equation (190"3) which 
within themselves the necessary and sufficient conditions that the ellipsoid 
and the elimination of 6 between this and the third equation gives 
(a 2 — b~) \a?b 2 J A jg— c 3 Jq\ = 0. 
(191-1), 
or by taking 
a 2 b 2 J ÀK — c 2 J 0 
(191-2). 
(192-1), 
* Fogg. Ann. xxxin. (1834), p. 229. 
(192-2).