378 PHILOSOPHICAL TRANSACTIONS. [aNNO 17/8. 



A prudent philosopher is always afraid to pronounce generally concerning the 

 existence of causes, which are attended with a variety of circumstances, and are 

 complex in their operations. To say that the quantity of force in bodies remains 

 invariably the same, seems to be a proposition of this kind. The mutual actions 

 of bodies on each other, especially when their gravity is taken into the question, 

 depends on so many considerations, and the cases which may be put are capable 

 of such an infinite variation, that it is impossible almost to draw a general infer- 

 ence of this nature. Even when experiments are produced, which seem to prove 

 the point, one is apt to suspect the universality of the conclusion, and to ima- 

 gine that it may possibly be owing to some particular circumstance which we have 

 not attended to, or been able to distinguish from others not so essential. In the 

 example we are considering, it is clearly proved from experience, that p X v'^ is 



equal to/jw* -\ ^; but whether that be true in every other case that may be 



conceived, can never be determined from such an experiment; nor is it possible 

 to make any distinctions about it, until we have demonstrated its connexion with 

 some other principle, which is more simple and less contested. 



Retaining the same symbols, let f represent the force of gravity, and/" the 

 I'orce which accelerates the body o. in its motion. From what has been already 

 shown it appears, that f:/:: o/j + bp:ap, and f — f:/:: l-p : np ■.■.'—^ • pv ; and 



because /)t' is the motion generated in a by the forcey, -^ will be the motion 

 lost in the same body a by the diminution of its gravity. Let a be any prismatic 

 particle of the body, and ad its distance from the axis; the velocity of this par- 

 ticle will be ^' ^ "^ ; its motion ; and, by the nature of the lever, the 



a (I "^ ' 



motion which a must lose to generate such an effect in a must be -^ "^f ^ ^. 



a* 



The quantity — ^ represents the sum of all the quantities ^ — ; and there- 

 fore the motion, which q has lost by its action on the body, is precisely equal to 

 the motion gained by the different parts of that body after a proper allowance is 

 made for the lengths of the levers, ad, &c. 



Thus it appears, that there is no necessity, in accounting for the time of q's 

 descent and the velocity it acquires, of having recourse to the conservatio vis 

 vivae, or any such perplexed hypothesis. By pursuing the analytic method far 

 enough, we have been led directly to that fundamental law of motion, that ac- 

 tion is equal to re-action, and in the contrary direction. 



A distinction however is always to be made between the actions of bodies when 

 at liberty, and when they revolve about a centre or axis. In the first case, the 

 motion lost is always equal to the motion communicated in an opposite direction: 

 in the second, the motion lost is to be increased or diminished in the ratio of the 



