1 34 Cataclysmic Motion [en. vi 



It seems likely that this furrow will deepen until the mass divides into 

 two parts. If so the motion, which must be in the plane JL in fig. 25, may 

 end by the representative point coming to rest at some point such as L on 

 one of the stable double-star figures which consist of two stars revolving 

 about one another. Thus the " cataclysm " which must occur when the repre- 

 sentative point reaches the configuration J may be represented by a jump 

 of this point from J, the highest stable configuration on the ellipsoidal series, 

 to L, a configuration of equal angular momentum on one of the double-star 

 series. 



Some of these double-star series do not possess stable configurations 

 having angular momentum equal to that of the critical Jacobian ellipsoid /. 

 Denoting this latter angular momentum by 0'3898, so as to conform to the 

 measurement of angular momentum used in our tables on pp. 39 and 40, I 

 find the following angular momenta for Darwin's figures of limiting partial 

 stability tabulated on p. 63*. 



M'lM 0-33 0-4 0-5 1-0 



Ang. Momentum = 0110 0'390 0'413 0'440 0-481 



Since all stable figures for which M'\M > 0'33 have angular momentum 

 greater than that of the critical Jacobian ellipsoid, it is evident that an 

 imcompressible mass cannot divide by fission into two masses more nearly 

 equal than 3:1. This theoretical upper limit of W/M for incompressible 

 masses is just about equal to the observed lower limit of M'jM for actual 

 masses (cf. 2), but compressibility may tend to equalise the ratio of the 

 masses. 



We have so far supposed that the two masses will assume a position 

 of relative rest, rotating as a rigid body. Other possibilities, such as that 

 of the masses describing non-circular orbits about one another or of the 

 periods of rotation and revolution not coinciding, ought also to be considered ; 

 for convenience this is deferred to Chap. XI. 



III. THE DOUBLE-STAR PROBLEM 



134. We found in 58 that in Roche's problem of an infinitesimal 

 satellite revolving in a circular orbit about a massive primary, there is a limit 

 of closest approach within which no stable configurations of equilibrium exist 

 for the satellite. Thus if a small satellite falls, or is in any way driven, into 

 a certain sphere surrounding its primary, its configuration will become un- 

 stable and dynamical or cataclysmic motion must ensue. We have further 

 seen in 60 65 that in the more general double-star problem a precisely 

 similar situation arises, and it will be clear that the dynamical motion in 



* The last figures cannot be guaranteed as I have assumed for 1 + f the uniform value 1-06 

 from M'/M = 0-4: to M'/M=l. The entry corresponding to 3/'/3/=0-33 is obtained by inter- 

 polation. 



