118 Cataclysmic Motion [CH. vi 



I. THE TIDAL PROBLEM 



118. In the tidal problem there is no point of bifurcation at the 

 stage at which instability sets in. We have seen that there is a series 

 of spheroidal configurations which are thoroughly stable for eccentricities 

 from to '882579, but are unstable beyond. The vibration for which insta- 

 bility sets in is one in which the figure remains spheroidal but its eccen- 

 tricity varies. 



We shall now find that when, as assumed in Chapter III, the tide gene- 

 rating potential reduces to the simple form 



there is a possible motion in which the boundary remains spheroidal through- 

 out ; the question of whether this motion is stable, as well as that of what 

 modifications are introduced when the tidal potential does not reduce to this 

 simple form, will be discussed later. 



119. Let us consider the possibility of the general ellipsoid 



being a boundary for the fluid mass when in motion. The rates of change 

 of a, b, c will be denoted by a, 6, c. At every stage of the motion we must 

 have 



abc = r 3 , 

 so that we necessarily have 



And on again differentiating with respect to the time we obtain 



a b c d 2 6 2 c 2 



The velocity-potential of the motion, if the fluid is still assumed incom- 

 pressible, is given by* 



..................... (345), 



a c 



this satisfying the requisite condition V 2 4> = in virtue of equation (348). 

 The velocity v at any point is accordingly 



* Lamb, Hydrodynamics (4th Edition), 110. 



