138 THE TIDAL PROBLEM. 



ous. When they have reached, or are in, that condition of steady motion of 

 slow rotation postulated in the investigation, they are undoubtedly always 

 strongly condensed toward their centers. The other, and probably more 

 important, one is that isolated celestial masses do not change their rates of 

 rotation except when they change their densities or distribution of densi- 

 ties. While the theoretical discussions to which reference has been made 

 regard the rate of rotation as the single variable parameter, in the actual 

 case there is a corresponding change in density. The importance of not 

 neglecting the latter is easily seen. 



Consider a slowly rotating homogeneous fluid having the form of a nearly 

 spherical oblate spheroid. The eccentricity of a meridian section depends 



2 



upon the quantity ^ , 2 , where a» is the angular rate of rotation, k^ the 



gravitational constant, and a the density of the mass. The eccentricity of 

 the axial section increases with the increase of this function, provided it does 

 not go beyond a certain maximum. Now suppose the mass contracts in such 

 a way as to remain homogeneous throughout, and so that it continues to 

 rotate as a solid spheroid of equilibrium. Because the moment of momentum 

 of an isolated mass is constant, the contraction implies an increase in a>, and 

 therefore, as far as this factor alone is concerned, an increase in the oblate- 

 ness of the mass. But the contraction also implies an increase in a, and 

 therefore, so far as this factor alone is concerned, a decrease in the oblate- 

 ness of the mass. That is, keeping the moment of momentum constant, as 

 the dynamical situation requires, we find the eccentricity acted upon by 

 two opposing factors. If, under the influence of these factors, the figure 

 should become less oblate, the fission theory would get no support from the 

 discussion; if it should get more oblate, the question is at what rate the 

 mass must rotate and to what extent the contraction must proceed before 

 there is a possibility of fission. This paper will be devoted to a brief discus- 

 sion of these questions. 



