The last chapter contained a discussion of the ellipsoidal configurations which can occur in the various problems we have had under consideration, and it was found possible to investigate their stability or instability subject to their remaining ellipsoidal. A configuration which is unstable when subject to an ellipsoidal constraint will of course remain unstable when this constraint is removed, but a configuration which is stable before the constraint is removed will not necessarily remain stable. We can only discuss whether such a configuration is stable or not when we have a complete knowledge of all configurations of equilibrium adjacent to the ellipsoidal configurations; we then know the positions of the various points of bifurcation on the ellipsoidal series, and the stability of this series is immediately determined.

A first condition for being able to discover configurations of equilibrium of any type is that we shall be able to write down the potential of the mass when in these configurations. Thus it appears that before being able to discuss in a general way the configurations of equilibrium adjacent to ellipsoidal configurations, we must be able to write down the potential of a distorted ellipsoid.

The method of ellipsoidal harmonics at once suggests itself. It has been used by Poincaré Darwin, and Schwarzschild to determine configurations of equilibrium adjacent to the equilibrium configurations. In this way the various points of bifurcation on the ellipsoidal series we have had under discussion are readily determined.