Researches in the Theory and Calculus of Operations. 241 



verse radius squared, ^; but if counted by equal angles in 



equal times, is proportional to the direct radius squared, r 2 

 (Lardxer, Young). 



To the radius r, the volume of the sphere filled by ema- 

 nation from a centre of force (the Sun), is measured by 

 |**r*, proportional to the cube of the radius. Of this ema- 

 nated force, there is distributed on the surface of the sphere 

 an amount measured by 4*r, proportional to the square of 

 the radius; while any great circle of the sphere holds an 

 amount measured by 2*r, proportional to the simple radius. 

 In the radiation, expansion or equable diffusion of a force 

 in space, the extension is to the intension as the radius r to 



its reciprocal r'= ^ (Herschel). So that the comparison of 



two spheres of radii r' and r, OA' and OA (fig. 33), with 

 reference to intension, gives the proportions 



. |*r' 3 : 4*r' 2 : 2*r : : ^r 2 : 4*H : %tt*\ 



whence the equations 



r — L_ = L, the cubed radii referring to the (mean) dis- 

 r 3 r 2 r' 



tances of the planets, 

 the squared radii to the times of revolution, and 

 the simple radii to the relation of inverse dis- 

 tances (Kepler). 

 In a sphere of radius /• (that of a planet in its orbit), the 

 whole amount of force emanated by the Sun is equal to 

 jrfr 3 , the volume of the sphere. Of this amount, that which 

 applies to the planet is equal to *r, the area of the diame- 

 tral circle of the sphere intersecting the place of the planet; 

 and the circumference of the great circle drawn through the 

 planet constitutes its mean orbit, is equal to 2*r, and applies 

 to the revolution of the planet. In Kepler's second law, the 

 times of revolution are counted in squares (resultant of a 

 force of the second degree) ; and when equal distances are 

 measured by equal times of description, the third law is ob- 

 tained : the squares of the times proportional to the cubes of 

 the (mean) distances. 

 Trans, vit] 31 



