revolving at small distances from their Primaries, 417 



(21) will enable us to derive from (28) an equation for the pres- 

 sure of a fluid column having its section equal to unity, and ex- 

 tending from the centre to either of the poles. Denoting by P' 

 its pressure at the centre. 



Now, in accordance with the condition of equilibrium expressed 

 in formula (9), make the values of P and P' in equations (21) 

 and (29) equal; substituting for B 2 its equal, A 2 (l— e 2 ), and 

 dividing by y£ 2 7rA, we obtain 



n Jl, 1 + e 2\ 2R 3 2(1 -OR* * 



(1-^log — -_J-_ = _L_-^+(l- e 2 ) 



/l 1-6 2 l + 6\ 



V^~-2^ l0 Sl^> 

 which after some reductions and transposition gives 



R 3 3(l-e 2 )/3-e 2 1 + 6 3\ ^ 



It appears from equation (11), and from the principles on which 



it has been deduced, that when the stability of the satellite ceases to 



doo 

 be possible, -j- is equal to nothing ; and accordingly equation (30) 



and its differential might enable us to determine the size of the 

 smallest orbit such a body could describe, and its deviation from 

 a sphere previous to the final dismemberment. To avoid, how- 

 ever, the difficulties of the resulting equation and the ambiguity 

 of its roots, it will be advisable to make the estimate from a com- 



R 3 



parison of the following values of — g-, corresponding to different 



degrees of excentricity, and calculated from formula (30). 



An inspection of this Table shows that the elongation of a 

 satellite increases to a most enormous extent with the disturb- 

 ance, and that the dismemberment is inevitable when the value 



