47-49] The Tidal Problem 45 



Obviously these two series of configurations correspond roughly to the 

 series of Maclaurin spheroids and Jacobian ellipsoids in the rotational 

 problem. 



49. Eliminating 6 from equations (79) and (80) and dividing out by the 

 factor b' 2 c 2 , we obtain 



2irpabc 



This does not give the value of /A on the spheroidal series, for on this 

 series the factor b 2 c 2 vanishes. It gives the value of p on the ellipsoidal 

 series, and shews that /j, is necessarily negative throughout this series. Since 

 fji is positive in the physical problem, it appears that these ellipsoidal con- 

 figurations cannot actually occur ; only the spheroidal series remains as a 

 physical possibility. 



To obtain the value of //, on the spheroidal series we may eliminate 

 between equitions (78) and (81) and put b = c. We find that the spheroids 

 corresponding to positive values of //. are prolate (a > c), and for these 



d\ 2c 2 



_ _ 



X) f (c 2 + X) c * + 2a * 

 l+e 



in which e is the eccentricity, given by e 2 = (a 2 c*)/a?. 



Equation (85) shews that //, = when e = 0, as it ought to be ; it also shews- 

 that fjb = Q when 6=1. On treating equation (85) numerically, it is found 

 that p continually increases from the value /u, = at e = 0, until e has the 

 value '88257.9, after which /* decreases down to the value p = at e = 1. The 

 maximum value of //, is '125504^. 



Clearly the configurations of equilibrium which correspond to positive 

 values of /u, form a diagram similar to fig. 4 (ii). It follows that all spheroids 

 for which e < '8826 are stable, while all others are unstable. So long as 

 fj, < *1255047rp, there are always two spheroidal configurations, one stable and 

 the other unstable ; when //, > 125504^, there are no spheroidal or ellipsoidal 

 configurations at all. 



When a tide-generating mass approaches the primary mass, //, may be 

 supposed to increase continually. We now see that if /j, changes slowly enough,. 

 the primary passes through a series of prolate spheroids of continually in- 

 creasing eccentricity, until /-t reaches the value 0-1255047jy>. After this value 

 of fj, is passed the problem becomes a dynamical one. We shall give the 

 solution of this dynamical problem in a later chapter. For the present we 



