ss-ss] Rotational and Double-star Figures 85 



finite terms, and the equations can only be solved by trial and error. The 

 calculations have been carried through by Sir G. Darwin*, and the- solution 

 he obtains is 



a = l-885827r , & = -814975r , c = '650659r (229), 



the corresponding value of tf/Z-rrp being '1419990. The shape of the 

 ellipsoid, together with the pear-shaped figure derived from it, is shewn in 

 fig. 14. 



Double-star Figures 



87. The two points which have just been determined enable us to fix 

 with fair approximation the curve given by equation (223), which is the locus 

 of all first points of bifurcation. 



For in fig. 15, which reproduces that part of fig. 7 in which the series 

 of ellipsoidal configurations lie, the two points just determined are repre- 

 sented by the points B f and B". These points are so near to one another 

 that we may regard the straight line B'B" 'as a sufficiently good approxi- 

 mation to the position of the locus in this part of the plane. 



The curved line SR"T" represents the locus of points at which the ellip- 

 soidal configurations were found in Chap. Ill to become unstable through 

 the angular momentum becoming a minimum. It is clear that the line B'B" 

 cannot cross into the area marked off by this line, so that all configurations 

 on the line B'B" must already have become unstable in the double-star 

 problem. 



88. The results which have been obtained can be now summarised, with 

 reference to fig. 15 (on the next page), as follows : 



In the tidal problem, only the part ST" of the spheroidal series is stable ; 

 the part T"T is unstable. The range T"B"\$ unstable through a spheroidal 

 displacement only, and the range beyond B" is additionally unstable through 

 a pear-shaped displacement. 



In the double-star problem, only configurations represented inside an area 

 such as SR"T"S are stable; all others are unstable. The configurations 

 inside the area SR"T"B"B'BS are unstable through ellipsoidal displace- 

 ments only, while the range beyond B'B" is additionally unstable through a 

 pear-shaped displacement. 



In the rotational problem, the range SBB' is stable, while the range B'J 

 is unstable through a pear-shaped displacement. 



* " On the pear-shaped figure of Equilibrium of a Eotating Mass of Liquid," Phil. Trans. 

 198 A (1901), p. 301, or Coll. Works, in, p. 288. I have verified that Darwin's solution, which 

 was obtained by harmonic analysis, satisfies my equations (218) to (221). See Phil. Trans. 

 215 A, p. 53. 



