CHAPTER IV 



THE GRAVITATIONAL POTENTIAL OF A 

 DISTORTED ELLIPSOID 



66. 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 sirbject 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 con- 

 figurations 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 dis- 

 cuss 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 Poincare*, ! Darwinf, and SchwarzschildJ 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 dis- 

 cussion are readily determined. 



After determining the position of points of bifurcation on the ellipsoidal 

 series, the next problem is that of determining whether the branch series 

 through these points are initially stable or unstable, and this, as we have seen, 

 demands a knowledge of the direction of curvature of these branch series at 

 the points of bifurcation. We can only discuss the curvature of this series 

 when the configurations on it are known as far as the second order terms of 

 a parameter e, which measures the displacement from the original ellipsoidal 

 configuration. Poincare has devised a method of using ellipsoidal harmonics 



* Acta Math. 7, p. 259, and Phil. Trans. 198 A, p. 333. 



t Coll. Works, Vol. in. papers 10, 11, 12 and 13. 



J Neue Annalen d. Sternwarte Miinchen, 3 (1897), p. 275, or Inaug. Dissert. Miinchen (1896). 



Phil. Trans. 198 A, p. 333. 



J. C. 5 



