46 Ellipsoidal Configurations of Equilibrium [CH. in 



notice that the critical stage at which this dynamical motion begins is deter- 

 mined by 



-j- = 0125504777), 

 or, since M f 7rpr 3 , it is determined by 



r/vi 



f ) r (86). 



When the secondary approaches to a distance less than this, there are no 

 configurations of equilibrium for the primary. When the secondary is at this 

 critical distance, the primary has the greatest eccentricity consistent with 

 stability. This is given by e = "882579 and the lengths of the semi -axes are 



a = I'65390r ; b = c = 77757r , 

 these lengths being very approximately in the ratio 17:8:8. 



III. THE DOUBLE-STAR PROBLEM 



50. We now proceed to the third problem that of two bodies rotating 

 round one another without any change of relative position. This problem 

 has been studied in detail by Roche* and Darwin "f. 



Let the two bodies be spoken of as primary and secondary, and let their 

 masses be M, M' respectively ; let the distance apart of their centres of gravity 

 be R, and let the angular velocity of rotation of the line joining them be ew. 



It will be sufficient to fix our attention on the conditions of equilibrium of 

 one of the two masses, say the primary. Let its centre of gravity be taken 

 as origin, let the line joining it to the centre of the secondary be axis of a?, 

 and let the plane in which the rotation takes place be that of xy. Then the 

 equation of the axis of rotation is 



x = M' fR ^ = () 



The problem may be reduced to a statical one (cf. 31) by supposing the 

 masses acted on by a field of force of potential 



or | or (x- + y g ) - , Rtfx + cons (87). 



* E. Roche, "La Figure d'une Masse fluide soumise a 1'attraction d'un Point eloigne." 

 Acad. de Montpellier (Sciences), i. (1850), p. 243. 



t G. H. Darwin, "On the Figure and Stability of a liquid Satellite.'" Phil. Trim*. 206 A 

 ^1906), p. 161, and Coll. Work*, in. p. 436. 



