2 Trans. Acad. Sci. of St. Louis. 



The mass of a spherical shell of density o and thickness 

 dll is 



^iroRhlR. 



The potential of the sphere upon the matter of the sur- 

 rounding shell of thickness dR> is therefore 



d r = 3 ttoR'AttoRWR ( 2 ) 



Now suppose we regard R as variable, and find the integral 

 of the successive elements of the potential of the sphere upon 

 itself, when the radius changes from to R. This will give 

 the potential of each succeeding sphere upon its surface, or in 

 the limit the potential of the sphere upon itself. 



M 



16tt 2 . 

 ) = -7T- IJIPdR. (3) 



1: 







As the attraction of a homogeneous spherical shell with re- 

 spect to points within is zero, we may disregard the action of 

 any layer upon the inclosed sphere, and consider merely the 

 action of the successive spheres upon their surfaces ; the re- 

 sult will be the potential of the sphere upon itself, and cor- 

 respond to the total energy given up by the particles in 

 condensing from infinity. Accordingly when a is constant, 

 we have 



R 



16 _ 





Hence the theorem : The potential of a homogeneous sphere 

 upon itself is equal to three-fifths of the square of the mass 

 divided by the radius. 



If masses are expressed in grammes instead of in astronom- 

 ical units, equation (4) must be multiplied by the gravitation 

 constant r, the value of which may be determined in terms 

 of any special case of attraction. Thus let g be the accelera- 



