122 Cataclysmic Motion [OH. vi 



it has departed far from its original spherical shape. Thus during the short 

 interval in which /x is finite, we have approximately e = and a = c. 



It follows, from equations (359) and (361), that at any instant during 



this interval, 



a c 



where r is the mean radius. Integrating through the interval in which /JL is 

 appreciable, say from to r, we obtain 



a c 3r I pelt, 



Jo 



where a and c are the velocities at the end of the interval in question. We 

 also have d.-f- 2c = from the condition of constancy of volume, so that 



-=-- = 2 ( T fidt (363). 



n r Jo 



Each fraction is equal to fee, so that these equations give the value of e, 

 and therefore the kinetic energy, at the instant t = r. The subsequent 

 motion is of course one under no applied forces, with this assigned amount 

 of energy, and so can be completely determined. 



The energy equation is readily found to be* 



|(3 _ <) (1 - #)-**# = 6 [//*]' - 4* P J2 - ( =/- log \ 

 and the turning points, at which e = 0, are accordingly determined by 



We may notice that if I pelt > ($7rp)* , there will be no turning point 



and the motion will overshoot the value e=l. But we shall immediately 

 see that such motions as this cannot occur, for instability will be set up 

 before the value e = 1 is reached. 



124. We have seen that a motion which satisfies the dynamical equations 

 is one in which the figure remains always spheroidal, the velocity potential 

 being given by 



(365) 



in which b and c remain equal. This motion has not however been shewn to 

 be stable. 



* For details, see Memoirs R.A.S. Vol. 72 (1917), p. 19. 



