412 Mr. D. Vaughan on the Form of Satellites 



Substituting for M and m their values, -^ — and — „— , this 



becomes 



4y*V(R 3 + r 3 )r 



3x* 



(G) 



Subtracting the sum of the expressions (2) and (6) from (1), 

 there results the equation 



in which F represents the force of gravity at that part of the 

 satellite in conjunction with the primary. If, instead of adhering 

 to the hypothesis I have adopted, we suppose the axis of rotation 

 to form an oblique angle with the plane of the orbit, F will be 

 variable, and formula (7) will express only the lowest limit which 

 it attains during every revolution. Were the satellite homoge- 

 neous and very inferior to the primary in magnitude, all the 

 matter between its centre and any point on its surface must lose 

 weight in nearly the same ratio by the disturbing force. 

 By making F in equation (7) equal to nothing, we obtain 



This value of x represents the radius of the circular orbit in which 

 a spherical satellite would be incapable of retaining disconnected 

 bodies at the place next the primary, and could give only a very 

 insignificant weight to the matter on a direct line between this 

 and the opposite part of its surface. In the case of the earth 

 and moon, the value of x in equation (8) would be a little over 

 7500 miles ; and were the lunar orb made to revolve in a circle 

 of so small a radius, gravity would disappear at the part of its 

 surface nearest to us, so that there could be no adequate coun- 

 terpoise to the enormous pressure arising from the weight of 

 matter in other localities. It therefore becomes necessary to 

 correct our hypothesis respecting the figure of a satellite in such 

 circumstances. A far less degree of proximity to our globe would 

 be sufficient to give the moon a very considerable distortion from 

 a true sphere ; and as their cohesive force, when once overcome, 

 could oppose little resistance to their arrangement, the lunar 

 materials would ultimately exhibit a form differing little from 

 that which a fluid might assume under the operation of the same 

 forces. 



In taking up the case in which the satellite is to be regarded 

 as fluid, I think it advisable to have recourse to an approximate 

 method of investigation, which may be conveniently extended 

 for the attainment of any desirable degree of accuracy, while it 



