140 



THE TIDAL PROBLEM. 



dm 



It is well known that -^V'^ passes through zero but once between .^ = and 



>l= 00. Hence we may represent in figure 17 the facts so far enunciated. 

 As 0) starts from zero and increases, there is a series of figures of equilibrium 

 starting from and another from P, the two series coinciding and vanishing 

 at the point a. 



Fig. 17. 



We may also indicate the existence of the Jacobian ellipsoids of equi- 

 librium on this figure, without, however, being able to define completely 

 their shape by a single point on a curve. Let us represent the eccentricities 

 of the sections made by planes passing through the axis of rotation and each 

 of the other two axes of the ellipsoid by abscissas in figure 17. The Jacobian 

 ellipsoids branch from the Maclaurin spheroids at h. For a given value of 



the corresponding point on the curve be gives the eccentricity of the 



section through the longest axis and the axis of rotation, while the cor- 

 responding point on hd gives the eccentricity of the section through the 

 remaining axis and the axis of rotation. These two points together com- 

 pletely define the shape of the ellipsoid. 



However, we shall regard the two series of ellipsoids he and hd as dis- 

 tinct, the properties indicated by a point on either of them being sufficient, 

 when taken with certain equations of relation not represented on the dia- 

 gram, completely to define the figure. That is, each curve of the whole 

 diagram will be regarded as carrying with it a certain set of equations which 

 serve to complete the definition of the shape of the figure of equilibrium 

 corresponding to each of its points. Thus, OaP carries with it the equation 

 which says that the eccentricity of every plane section through the axis is e. 



