VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 467 



rectangle rc contained between the lines rd and cd. From hence we may draw 

 the following corollaries. 



8. Carol. 1. The central body def remaining the same, and consequently the 

 forces at the same distances remaining also the same, the areas of the rectangles 

 rc, rc will always be inversely as the distances of the points d, d from c, their 

 sides rd, rd being inversely in the duplicate ratio of the sides cd, cd: and 

 therefore, because the velocity of a body falling from an infinite height towards 

 the point c, is always in the sub-duplicate ratio of these rectangles, it will be in 

 the sub-duplicate ratio of the lines cd, cd inversely. Accordingly, the velocities 

 of comets revolving in parabolic orbits are always in the sub-duplicate ratio of 

 their distances from the sun inversely ; and the velocities of the planets, at their 

 mean distances (being always in a given ratio to the velocity of such comets, viz. 

 in the sub-duplicate ratio of 1 to 2) must necessarily observe the same law also. 



9. Cor. 2. The magnitude of the central body remaining the same, the velo- 

 city of a body falling towards it from an infinite height, will always be, at the 

 same distance from the point c, taken any where without the central body, in the 

 sub-duplicate ratio of its density ; for in this case the distance cd will remain the 

 same, the line rd only being increased or diminished in the proportion of the 

 density, and the rectangle rc consequently increased or diminished in the same 

 proportion. 



10. Cor. 3. The density of the central body remaining the same, the velocity 

 of' a body falling towards it from an infinite height will always be as its semi- 

 diameter, when it arrives at the same proportional distance from the point c : 

 for the weights, at the surfaces of different spheres of the same density, are as 

 their respective semi-diameters ; and therefore the sides rd and cd, or any other 

 sides rd and cd, which are in a given ratio to those semi-diameters, being both 

 increased or diminished in the same proportion, the rectangles rc or rc will be 

 increased or diminished in the duplicate ratio of the semi-diameter cd, and con- 

 sequently the velocity in the simple ratio of cd. 



11. Cor. A. If the velocity of a body falling from an infinite height, towards 

 different central bodies, be the same, when it arrives at their surfaces, the den- 

 sity of those central bodies must be in the duplicate ratio of their semi-diameters 

 inversely ; for by the last cor. the density of the central body remaining the 

 same, the rectangle rc will be in the duplicate ratio of cd ; in order therefore 

 that the rectangle rc may always remain the same, the line rd must be inversely 

 as cd, and consequently the density inversely as the square of cd. 



12. Cor. 5. Hence the quantity of matter contained in those bodies must be 

 in the simple ratio of their semi-diameters directly ; for the quantity of matter 

 being always in a ratio compounded of the simple ratio of the density, and the 

 triplicate ratio of their semi-diameters, if the density be in the inverse duplicate 



3 o 2 



