APPLICATIONS TO THE SOLAR SYSTEM. 151 



meridian section of a homogeneous fluid having the mass and the mean 

 density of Saturn, and rotating in its period, would be 0.607. Since the 

 constitution of the sun is in a general way like that of Saturn, it will have 

 the oblateness of Saturn about when the section of the corresponding 

 homogeneous mass has this eccentricity. Making the computation from 

 equation (6) we find that the sun will not become so oblate as Saturn is now 

 until its mean density becomes 148 X 10^° on the water standard. This cor- 

 responds to a radius of 37.3 miles. If we may regard this as an impossibly 

 large density we may conclude that the sun will never be so oblate as Saturn 

 is now, and that its stabiHty will always be greater than that of Saturn at 

 present. 



Apparently the chances that Saturn will separate into two parts because 

 of shrinking and rapid rotation are greater than that any other member of 

 the solar system will ever suffer fission. To examine the probabilities we 

 shall apply equations (4), (5), and (6) to Saturn. Taking the density, mass, 

 and period of rotation as 0.72, 37V2J and 10.25 hours respectively, and com- 

 puting the moment of momentum under the hypothesis that Saturn is now 

 a homogeneous sphere, in order to give the theory that fission is possible all 

 the benefits of the approximations, we find that when Saturn shall have an 

 oblateness equal to that of the spheroids from which the Jacobian ellipsoids 

 branch, its density will be 21 times that of water, its axial diameter 16,500 

 miles, its equatorial diameter 28,400 miles, and its period of rotation 1 hr. 

 24 m. The high mean density demanded seems to be fatal to the theory of 

 fission in this case. 



In order to see how great changes in the density, dimensions, and period 

 the body will undergo by the time it reaches the state where the pear- 

 shaped figures branch, we may apply equations (7), (9), and (10) to Saturn. 

 Darwin has made the computations^ from equations equivalent to (7) and 

 (8), and has found that for this point 



/I = 0.7544 A' = 2.7206 2^^ = ^.1420 



Then equations (9) show that at this stage the mean density of Saturn 

 must be 93 on the water standard, its polar diameter 9,400 miles, its longest 

 diameter 27,000 miles, and its period of rotation 46 minutes. That is, the 

 mass is about four and one-half times as dense when the pear-shaped figures 

 branch as when the Jacobian ellipsoids branch. While the computation 

 was apphed to Saturn, it follows from equations (6) and (10) that this same 

 ratio for the densities at these critical forms is true, whatever the mass and 

 moment of momentum of the body under consideration. 



The density which the earth will attain before it will reach one of the 

 critical forms is so great that the computation is without interest. But we 

 may examine the hypothesis that the earth and moon were originally joined 

 in one mass whose rapid rotation produced instability, and that resulting 

 fission gave rise to two bodies having great stability. It is to be observed 

 in the first place that the moment of momentum of the earth-moon system 

 has remained constant except for influences exterior to itself. There is none 



^ Phil. Trans., A, 198 (1902), p. 326. 



