SUMMARY, 159 



and the moment of momentum. When the oblateness of the spheroid is 

 given there is but a single density satisfying the conditions, and when the 

 density is given there is but one spheroid satisfying the conditions. 



For the applications we assume that an actual celestial body will not be 

 in danger of fission until the corresponding homogeneous incompressible 

 body arrives at the state where the Jacobian ellipsoids branch. The density 

 at this stage is less than one-fourth that at which the pear-shaped figures 

 branch, and actual fission in the homogeneous bodies is certainly beyond 

 this form, if indeed fission into only two bodies is ever possible. With this 

 very conservative assumption we proceed to some calculations. 



(1) We find that the sun can not arrive at this critical stage until its 

 mean density shall have exceeded 307 X 10*^ on the water standard. This 

 corresponds to an equatorial diameter of the sun of about 22 miles. 



(2) We find that the sun can not become so oblate as Saturn is now until 

 its mean density shall have exceeded 148 X lO^" on the water standard. This 

 corresponds to an equatorial diameter of the sun of about 75 miles. 



Since even the latter density is impossibly great we conclude that the 

 sun will never become so oblate as Saturn is now, and that it will always be 

 more stable than Saturn is now. 



(3) We find that Saturn can not arrive at the critical stage at which the 

 Jacobian ellipsoids branch until its mean density shall have become 21 

 times that of water. This corresponds to a polar diameter of 16,500 miles 

 and an equatorial diameter of 28,400 miles. We conclude because of the 

 great density demanded that Saturn will never suffer fission. 



(4) We assume that the earth and moon were once one mass and get 

 their original moment of momentum from its present value. In computing 

 it, however, we make certain approximations so as to get it too large and 

 thus favor the conclusion of fission, then we add to it the maximum amount 

 the sun's tides can have taken from the earth, and finally we add 25 per cent 

 for fear there may be some unknown sensible factors omitted. Then we 

 find that this hypothetical earth-moon mass could not get even to the 

 critical point where the Jacobian ellipsoids branch until its mean density 

 became 215 times that of water, or about 40 times the present mean density 

 of the earth and moon. It would not become even so oblate as Saturn is 

 now until its density had become 10.4 times that of water. Therefore we 

 conclude that the hypothetical case was false, and that the moon has not 

 originated by fission from the earth in this way. 



(5) In applications to the binary stars the results are less definite because 

 of the meager data regarding these systems. But assuming that fission in 

 stars will occur when the Jacobian ellipsoids branch in the corresponding 

 homogeneous masses, we find for the density o in terms of water at the time 

 of fission when the two stars are of equal mass 



0.016 

 ^<— p2- 



where P must be expressed in mean solar days. Even though fission should 

 not occur until the density is ten times this amount (which, if true, makes 

 the evidence against fission in the solar system much stronger), all visual 



