CHAPTER III 



ELLIPSOIDAL CONFIGURATIONS OF EQUILIBRIUM 



33. The best-known configurations of equilibrium of a rotating homo- 

 geneous mass, namely Maclaurin's spheroids and Jacobi's ellipsoids, are both 

 of the ellipsoidal form, and this form will prove to be of primary importance 

 in all the cosmogonical problems we shall attempt to solve. We accordingly 

 devote a chapter to the subject of ellipsoidal configurations. 



Looked at merely from the point of view of convenience in the develop- 

 ment of the subject, the ellipsoidal form has the advantage that the potential 

 of an ellipsoidal mass is known and is comparatively simple, and that the 

 ellipsoidal configurations provide admirably clear examples of Poincare's 

 theory of linear series and stability. These reasons alone might justify our 

 studying ellipsoidal configurations in some detail, but there are weightier 

 reasons, as we shall soon see. 



Throughout this chapter and the three succeeding chapters the matter 

 under discussion will be supposed homogeneous and incompressible ; the more 

 complicated problems presented by non- homogeneous and compressible masses 

 will be attacked in Chapter VII. 



We shall deal in turn with three distinct problems the first, that of a 

 mass of liquid rotating freely under its own gravitational forces ; the second, 

 that of a mass devoid of rotation but acted on tidally by another mass ; the 

 third that of two masses rotating round one another and acting tidally on 

 / one another. The ^rst__groblem is of course-of. inie^eaL.in-J2Qnn-ectioji with 

 the rotational theory of pla^ietary__evojutipri_^ the second is of interest in 

 connection with the tidal theory ; while the third isjDf interest as -linking-iip 

 the two TolHineP~probleTns, ^ind also in connection with soi]oe_double-star 

 problems: fir-every^ene of Tbese prtrblemspwe^Iiall find ultimately that 

 the only stable configurations are of the ellipsoidal form, or are ellipsoids 

 slightly distorted by tidal inequalities. 



34. Notation. When one ellipsoid only is concerned, we shall take a, 6, c 

 to be its semi-axes, so that the equation of its boundary will be 



x 2 y' 2 z 2 



a' + l+c^ 1 < ol > 



As in many ellipsoidal problems, it will be convenient to think of the 

 surface (51) as being the surface X = in the family of confocal ellipsoids 



(52) ' 



32 



