60 Ellipsoidal Configurations of Equilibrium [CH. m 



but the value of R is now given in terms of &> by 



R = (M+M')*(l + tfe>~* ..................... (128), 



so that M, expressed in terms of &> alone, becomes 



M = (Mk* + M'k'*) 



(M+M'Y 



This differs from the former value of M obtained in 55 only through 

 the occurrence of the factor (1 + )*, and as this is never far from unity, it is 

 clear that ;the general discussion of stability given in 56 and 57 will remain 

 valid, at least in its general features. Always, except in the special case of 

 p = oo , there is a configuration in which M is minimum ; starting out of 

 this are two series of configurations along each of which M increases in- 

 definitely up to M = oo , these end configurations each being configurations 

 of zero rotation (ro = 0). One of these series ends in two spherical masses 

 rotating infinitely slowly round one another at an infinite distance, and this 

 series is stable throughout. At the end of the other series the primary 

 is an infinitely elongated Jacobian ellipsoid, and this series is unstable 

 throughout. 



Instead of eliminating R from equations (127) and (128), and so obtain- 

 ing M as a function of o>, we might equally well have eliminated co from 

 these equations and obtained M as a function of R. The equation obtained 

 in this way is 



M = \M& + M'k'* + R*} (1 + ?) 4 ( M + M '$ R ' * 



When M is not very different from M' , the value of M reduces to its last 

 term when R is large. Even for configurations in which the ellipsoids are 

 almost in contact, it is readily seen that by far the greater part of the value 

 of M comes fro'in this last term, so that M varies approximately as &. It 

 follows that the configuration for which M is a minimum nearly coincides 

 with that for which R is a minimum, this latter being the configuration of 

 closest approach of the centres of the ellipsoids. 



Let R be the distance of closest approach. Then for any value R + BR 

 which is just greater than R there will be two configurations ; in one the 

 ellipsoids are more elongated than in the configuration of closest approach, 

 while in the other they are less elongated. In the former configuration, 

 then, k z , k* and f are all greater than in the latter, so that, as the values of 

 R are the same in both configurations, it follows that the more elongated 

 configuration has the larger value of M. 



Now the diagram of configurations, drawn with M as vertical coordinate, 

 lies as in figure 8. Here is the configuration of minimum angular mo- 

 mentum, the less elongated configurations OP are stable, while the more' 



