﻿514 Mr. D. Vaughan on Secondary Planets 



for N in equation (6) ; there results 



Smk 2 mk 2 2. ON 



~2A == 'B' > nce 3 ( ' 



From equations (8) and (12) it will also appear, after some re- 

 ductions, that 



C C 3 0,4c 3 



B =1 ""CA 3 ° r B =1 ~8lB 3J * ' * (13) 



from which C may be found equal to -9576B or to '6384 A. 

 Though not a true ellipsoid, the satellite may, without much 



error, be considered equal in volume to ~ or — » — x 4256 ; 



o o 



and if p denote its mean density, and g the attractive force of a 



unit of matter at the distance k } then 



m= **M* x4256 , 

 o 



Regarding the primary as an exact sphere, its radius being r, and 

 its density being taken as unity, M (the measure of its attraction) 

 will be given by the formula 



477^ 



3 



These values of M and m being substituted for them in equation 

 (12), there is found, on reducing, 



D 3 7-049 _ l-9l7r ' 



- T = or D= 3/ — (14) 



This determines the size of the smallest orbit in which the sa- 

 tellite could preserve its integrity. If revolving a small distance 

 beyond the surface of the great central orb, it could not hold its 

 parts together unless its mean density were more than seven 

 times that of the primary. To the constitution which I have 

 assigned to the satellite many comets seem to approximate ; and 

 from the degree of compression which these bodies exhibit, or 

 from their power to resist the dismembering action of the sun at 

 certain distances from him, we may hope to gain some informa- 

 tion of their masses and densities by means of formulae similar 

 to those I have deduced. But the investigation for comets 

 could not have so well- denned a basis, nor lead to very certain 

 results, as the shape which their attenuated matter assumes de- 

 pends not on gravity alone, but on forces which seem removed 

 from the domain of scientific inquiry. 



For some of the questions involved in the case of a homoge- 

 neous satellite, an approximate solution was given in my com- 

 munications to the Philosophical Magazine for December 1860 



