332 APPENDIX. A. 



distance of the particles, at the surface of the sphere of ac- 

 tion, from its centre. But the mean distance of each ele- 

 mentary pyramid from its vertex, or of the whole sphere 

 from the centre, is f of the height or the radius, since the 

 products of the elements of the content into the distance 



added too^ether and divided by the content, or — ^- 



=1^. The mean variation of distance for the whole fluid 



is therefore ^ c — ; and this variation, multiplied by the 



number of particles within the sphere of action, becomes 



_ 5rc'* — ; which being again multiplied by the number of 



centres Mud, and by the force/, and divided^by du, gives 



us F=:^ tZ^c* Mf, for the whole force acting on the sur- 

 o 



face M. 



Corollary. In this case if the two forces C and R 

 hold each other in equilibrium, we must have c*C=^r*R, 

 and C must be to R, for each pair of particles, as r^ to c* : 

 each force still varying as the square of the density. 



Scholium 1. The determination of the attractive or 

 repulsive force of a sphere thus constituted may be illus- 

 trated and confirmed by a simpler mode of considering the 

 joint action of the particles of each hemisphere, which is 

 easily shown to be half as great as if they were collected 

 into one line. For it is obvious that each particle in any 

 spherical surface must have its action on the central point 

 reduced in the proportion that the radius bears to its dis- 

 tance from the plane dividing the hemispheres, conse- 

 quently the whole force will be represented by the distance 

 of the centre of gravity of the surface, multiplied into the 



