124, 125] The Tidal Problem 125 



same as if the figure had remained strictly spheroidal, so that the motion 

 may be supposed to be accompanied by an increase in the eccentricity e 

 until a turning point is reached at which e = and a = b c = 0. At this 

 point equation (376) assumes the simple form 



............ (377), 



since we have seen that 6' becomes identical with 6 at the turning point. 



The stability or instability of the motion at the turning point depends 

 on the signs of d 2 q l jdt 2 ) and since the factor s l is, by its definition, always 

 positive, these signs are those of the right-hand members of equations such 

 as (377). But the right-hand members of equations (377) are exactly the 

 quantities of which the vanishing determines the points of bifurcation on 

 the spheroidal series. They are all negative so long as no point of bifur- 

 cation has been passed on the spheroidal series, and one of them changes 

 sign at each point of bifurcation. 



The first point of bifurcation, as we have seen ( 85), occurs when 

 e = -947741, and corresponds to a third, harmonic deformation. If the 

 turning point, at which e = 0, occurs before e has reached the value "947741, 

 then the right-hand members of all the equations such as (377) will be 

 negative at the turning point, so that the dynamical motion will be stable 

 up to the turning point and also in returning, and the mass will sink back 

 into a spherical configuration. 



But if the turning point occurs just after e has reached the value '947741, 

 the dynamical motion at the turning point will be unstable through a third 

 harmonic displacement. Thus after passing the point of bifurcation a third 

 harmonic displacement will appear and will increase very rapidly, at least 

 until after the eccentricity has again diminished to below '947741, at which 

 stage the third harmonic vibration will again become stable, so that what- 

 ever third harmonic displacement there may be will oscillate and finally 

 disappear. The condition that the mass shall depart from the spheroidal 

 form is thus seen to be that the turning point shall occur for a value of e 

 greater than '947741. 



The intensity of tidal action necessary for this to occur can be determined 

 accurately in two cases (i) when //, changes very slowly, (ii) when p changes 

 so rapidly that the tidal action may be treated as " impulsive." 



When fju changes very slowly, any value of //, greater than '125504 TT^ 

 will suffice to set up dynamical motion and e will continue to increase until 

 after /n has again receded below the value '125504 wp. Hence when //- changes 

 very slowly the eccentricity will pass above '947741 if fju exceeds '125504 TT/O. 

 In this //, stands for M'/R S , where M' is the mass of the tide-raising body 



