132, 133] The Rotational Problem 133 



two separate masses just in contact. This configuration belongs not only to 

 the pear-shaped series of figures but also to one of the double-star figures 

 investigated by Darwin ( 60 65). It forms one of a series of figures PQR 

 of which the remaining members consist of double-stars rotating at different 

 distances apart. If Q is the figure of least angular momentunrrw^ know that 

 configurations on the branch PQ including P are unstable, while configura- 

 tions on the other branch QR are stable. Besides the series PQR, which is 

 continuous with the pear-shaped series, there are an infinite number of other 

 series of double-star figures, corresponding to all possible values of the ratio 

 of the masses. 



133. In the light of this knowledge we may examine what motion is to 

 be expected in a Jacobian ellipsoid which has reached the point at which 

 secular instability sets in. 



In fig. 26 let JJ' represent the series of stable Jacobian ellipsoids in the 

 neighbourhood of the point of bifurcation J. For any configuration within 

 the range //', the third harmonic 

 (pear-shaped) vibration is stable both 

 ordinarily and secularly. Thus if any ^ 



small pear-shaped vibration is set up / 



when the mass is in a configuration 

 such as A y the representative point j 



\ 

 \ 

 / 



B" \ D 



A" 



will oscillate backwards and forwards ' 



through some small range such as ' 



A A A" until the vibration is damped - 



by viscosity. If the vibration is set 



up when the representative point is 



at some point B close to /, there may 



Fig. 26. 

 still be oscillation through a small 



range, but the motion can only be stable if this range is less than the range 

 B'B" in fig. 26. For the point B" represents a secularly unstable configura- 

 tion, so that if the representative point once passes beyond B", on the 

 line BB"D, it will not return but will describe some path such as BB"D 

 in the plane through B. 



As the point B approaches J the range of vibration which is possible 

 without instability setting in becomes smaller and smaller and finally vanishes 

 altogether, so that in the limit any disturbance, no matter how slight, causes 

 the representative point to move permanently away from the line J'J. The 

 path of this point is necessarily in the horizontal plane through J, and we 

 know that the direction of this path initially is that of the tangent JL at J 

 to the pear-shaped series JB". In other words the motion is one in which a 

 furrow first forms on the ellipsoid, as in fig. 14, and this furrow continually 

 deepens. 



