FIGURES FOR HETEROGENEOUS FLUIDS. 



145 



IV. FIGURES OF EQUILIBRIUM OF ROTATING 

 HETEROGENEOUS FLUIDS. 



We now enter a field beset with formidable difficulties and in which 

 there are but few positive results. We shall consider masses which are 

 possessed of slow rotation, which grow continually denser toward their 

 centers, and which are approximately spherical in form. To simplify the 

 problem we may suppose first that we have under consideration a body 

 composed of a number of incompressible fluids of different densities, arranged 

 in order of increasing density from the surface to the center. This hypoth- 

 esis more nearly agrees with the conditions found in nature than that of 

 homogeneity does, and the results obtained under it may be taken as throw- 

 ing light on the actual problems. 



Clairaut has shown* that such a body as we are considering will always 

 be less oblate than it would if its mass were uniformly distributed through- 

 out its volume. This is easy to see in considering the limiting case of a 

 dense nucleus surrounded by a homogeneous atmosphere of vanishing mass. 

 That this result may be legitimately applied to the celestial bodies is proved 

 by the fact that the earth, Jupiter, and Saturn are all less oblate than they 

 would be if they were homogeneous and rotating at their respective rates. 

 It is an interesting and important fact that the differences in oblateness 

 of these planets and the corresponding homogeneous figures of equilibrium 

 are greater the smaller the mean density. That is, if a low mean density 

 means the mass is largely gaseous and compressible, we may conclude that 

 the more a body is condensed toward its center the less oblate it will be for 

 a given rate of rotation. The facts for the earth, Jupiter, and Saturn are 

 given in the following table, where e has been computed from equation (1). 



Consider still the case of a heterogeneous incompressible fluid mass with 

 greatest density at its center. Suppose the density is given by an equa- 

 tion of the form , /o\ 



(; = (To+£(Ti (2) 



where cr^ is the mean density of the whole mass, and where Oi is always 

 finite and may be a continuous function of s. For £ = certain series of 

 figures of equilibrium are represented in figure 17. Let us suppose there is a 

 third axis in the figure perpendicular to the e and w-axes. We shall mark 

 off values of e along this axis. 



Let the conditions for a figure of equilibrium be 



Fi{xi, . . ., rc„, w, £)=0, ^ = l, . . ., n 



(3) 



See Tisserand's M^canique Celeste, 2, Chap. XIII. 



