156 THE TIDAL PROBLEM. 



vanishes for no other value of [i. Consequently the right member of (27) 

 has a single minimum at f^L= 1. But for other values of // the inequality of 

 (27) diflfers more from an equality, and hence we are not certain that /i = 1 

 gives the least value of a for a given P and k. But in many stars // is 

 undoubtedly near enough unity to make equation (27) useful. 



We shall suppose n=l and compute a for the value of X for which ap- 

 parently there is first any danger of fission. If we suppose that this first 

 occurs for that value of X for which the Jacobian ellipsoids branch, that is 

 for ^ = 1.395 . . . , we find, taking the units so that o will be expressed in 

 terms of the density of water when P is expressed in terms of mean solar 



''"'' .< "•"'« r28> 



<T<— p^— (28) 



Or, if the fission occurred when the pear-shaped figures branched from the 

 Jacobian ellipsoid, we find similarly 



0.071 

 ^<-pr- (29) 



Consequently, this discussion leads to the conclusion that in all binary sys- 

 tems in which the two masses are approximately equal, and in which the periods 

 are at least several years, as they are in the visual pairs, the fission must have 

 occurred, if at all, while the parent mass was yet in the nebulous state. The 

 data regarding binary systems as a class are so meager that probably no 

 stronger conclusion than this can be drawn from this line of argument. 



There is, of course, no a priori objection to the theory that binaries as a 

 class have originated by fission in the nebulous state. But there are at least 

 two rather distinct hypotheses as to how and why such fission may have 

 taken place. The first is that in the origin of a nebulous mass the factors 

 which have determined its initial condition may have brought it into exist- 

 ence with at least two nuclei of condensation whose magnitude and density 

 were sufficient to have led to a binary, even though the moment of momen- 

 tum may have been so low that, if the mass had been spheroidal with the 

 same mean density, it would have been a stable figure. Fission in this type 

 of masses is not under consideration here. The second is that the mass in 

 its earliest nebulous stage was in an approximately spheroidal form, densest 

 at its center with density decreasing outward through approximately sphe- 

 roidal layers, and that as a consequence of its high moment of momentum 

 it lost its stability and divided into two masses. This is the type of fission 

 under consideration here. 



Suppose a nebula of this latter type suffers fission. At the time of fission 

 all parts are rotating at the same angular rate, and one of the two parts 

 must have a mean density less than, or at the most equal to, the mean den- 

 sity of the original mass. Consequently one of the two fragments because of 

 its lower density and equal rotation, must have at least as great a tendency 

 to fission as that which led to the division of the initial mass, unless either 

 its form is one of greater stability, or the tidal forces of the other member of 

 the pair tend to keep it from breaking up. If, as seems probable, the ap- 

 proximate spheroid is the most stable figure of equilibrium, and if the mass 



