54 Ellipsoidal Configurations of Equilibrium [CH. in 



The minimum value of M now coincides with the maximum value of o>. 

 The series of configurations are those represented on the series (p = 0) in 

 fig. 7, and the minimum value of co occurs at the point R". Thus con- 

 figurations on the branch SR" are stable, while those beyond R" are 

 unstable. 



The actual value of o> at the point R" is given by 



&> 2 /27T,o = 0-04503*. 

 The general value of a> z /27rp on this series is 



_- M ' = *(?-YjL 

 2-n-p ZjrpR 3 3 \p) R* 



so that in the critical configuration we have 



= 2-4554^-) r ' (105). 



Thus a small satellite rotating about a rigid primary of mass enormously 

 greater than its own cannot be in equilibrium in any configuration whatever 



if its distance from the centre of its planet is less than 2*4554 (p'/p)* radii 

 of the planet. This distance is commonly spoken of as Roche's limit. 



The critical value of R may also be put in the form 



r (106). 



This may be compared with equation (86) which determined the limit of 

 closest approach under the tidal forces from a secondary mass when the 

 bodies were not in rotation. The critical value of R just found is about 

 twelve per cent, greater than that found in the former problem ; the difference 

 of course represents the disturbing effect of rotation on the primary. 



59. In the more general case in which M is not .infinitesimal, and the 

 angular momentum is given by the complete equation (104), the maxim inn 

 value for co is not so easily found since k 2 will vary with o>. It is however 

 clear that M will be infinite when co 0, and that co will again increase to a 

 maximum and again decrease, so that M will pass through a minimum value 

 which will again divide stable from unstable configurations. Again there 

 will be a limiting value of R similar to Roche's limit, and there will be no 

 configurations of equilibrium at all for smaller values of R than this. 



* Roche gave 0-046; both here and in equation (105) I quote the more accurate values 

 deduced from Darwin's calculations. (Coll. Works, in. p. 436.) 



