48 Ellipsoidal Configurations of Equilibrium [CH. in 



It is convenient to put MjM' =p, so that 



(94). 

 The equations then reduce to 



^_R)^ ........................... (95), 



27rpabc a 2 



and the special cases are now obtained on putting p = oo (the rotational pro- 

 blem) and 2> = - 1 (the tidal problem). 



Eliminating 6 from equations (96) and (97) we obtain 



while similarly the elimination of from (95) and (97) yields 



(a 2 - c 2 ) f" 



7 Jo (a 2 



~~r A o- ...... 



+ X) (c 2 + X) A 2-Trpabc 



These two equations are identical, except for differences of notation, with 

 the two equations which Roche takes as the basis of his discussion*. 



53. If we now remove c from equations (98) and (99) by the substitution 

 c = rj/ab, the resulting equations will give a, b in terms of /z and p. 



In these equations the value p = oo has already been fully discussed, and 

 found to give the series of Maclaurin spheroids and Jacobiari ellipsoids. For 

 all other values of p, the value p = leads at once to a=b = c, and so gives a 

 spherical configuration. 



For values of /u, other than //, = 0, the elimination of fj, from equations (98) 

 and (99) leads to an equation which gives p uniquely in terms of a and 6, and 

 either equation then gives yu, uniquely. Thus all solutions of equations (98) 

 and (99) may be represented on a graph in which a and b, both necessarily 

 positive, are taken as abscissa and ordinate. Each point in this diagram will 

 correspond to one and only one value of p and /JL. On drawing the loci 

 p = constant, we obtain the various linear series corresponding to different 

 values of jo or M/M' in Roche's problem. 



Since the value of p for any value of a and b has been seen to be unique, 

 it follows that no two of these linear series corresponding to different values 

 of p can ever intersect. The median line a = b is already occupied by the 



* Equations (4) and (5) (p. 247) of Koche's memoir. 



