revolving at small distances from their Primaries. 413 



serves to remove a peculiar difficulty attending the solution of 

 problems of this character. In his investigations respecting the 

 stability of a fluid planet as dense as the earth, Laplace finds 

 that a rotation performed in less than '10090 of a day would be 

 incompatible with its existence in a spheroidal figure ; but as its 

 equatorial gravity is only partially neutralized by centrifugal 

 force, he concludes that such a rapidly rotating mass would 

 remain undivided under some form which analysis does not 

 reveal. A complete cessation of gravity in certain directions is 

 not, however, essential to an unstable equilibrium, either in the 

 case considered by Laplace, nor in the one now under considera- 

 tion, which seems to be better suited for the removal of the diffi- 

 culties of questions of this nature, as the time of rotation is pre- 

 vented from changing with the alterations in the figure of the 

 body. The stability of a fluid satellite, consigned to a small 

 orbit and elongated in the direction of its primary, is evidently 

 at an end — not when gravity is wholly suppressed at the parts 

 through which the greatest diameter passes, but when the force 

 with which a column of matter coincident with this line presses 

 to the centre ceases to acquire, from a further elongation of the 

 body, any preponderance over the pressure of a similar column 

 of matter extending from the centre to the nearest point of the 

 surface. Thus putting P and P' for the pressures which the cen- 

 tral region sustains from two similar columns of matter extend- 

 ing along the greatest and the least diameters, and e for the 

 excentricity of the section made by a plane passing through both 

 and supposed to be an ellipse, the distance from the primary 

 when dismemberment must occur, and the greatest deviation 

 from a sphere, may be determined from the equations 



P-P'=0, (9) 



£-? = (10) 



de de 



Another method of arriving at the same result will appear when 

 we consider that a satellite, in this critical situation, must have 

 its ellipticity increased to an enormous extent by a slight increase 

 of the disturbance or by a small diminution of the distance ; and 

 instead of (10) we may have recourse to the equations 



I = infinity, or J=0, . . . (11) 



where x, as in the previous notation, represents the radius of the 

 orbit. 



Were the satellite small compared with the central sphere, it 

 could not deviate much in form from an ellipsoid, the minor and 

 mean axes of which differ by a quantity much less than the 



