VOL. LXVIII.J PHILOSOPHICAL TRANSACTIONS. 377 



dicular to the same plane. Let e represent the sum of all the particles multi- 

 plied by the squares of their respective distances from the axis; then e shall be 

 equal to p X ab, as is demonstrated by all the writers who treat of the centre of 

 gyration. Let v be the velocity actually acquired by q after it has descended 

 through the space r; v the velocity which it would have acquired by the same 

 descent, provided the body had fallen freely by its gravity; and because the vires 



vivae are incapable of diminution or increase, we have pV^ = pv" -\ — . For 



since v is the velocity of a at a certain period of its descent, and is to the velo- 

 city of any prismatic particle in the body, as the distance mD from the axis to the 



Pool) 



distance of that particle from the same, it is evident that — :— will truly repre- 

 sent the sum of all the particles multiphed by the squares of their velocities, v^ 

 is therefore to i-- as ap + b? to ap, and the whole force of gravity is to the force 

 which accelerates the motion of a in the same ratio, because in uniformly acce- 

 lerated motions, when the spaces described are the same, the accelerating forces 

 are in the duplicate ratio of tiie velocities. It is obvious that the motion of q is 

 uniformly accelerated, because the velocity acquired by any descent, is to the 

 velocity of any point in the body, always in the same ratio; and therefore the 

 action of q on the body is the same as if both were at rest. Further, the alti- 

 tude z, through which a heavy body must fall to acquire the velocity v, is plainly 

 equal to r X — ~,~ ; for the altitudes z and r are inversely as the forces which 



^ ap + hp ■' 



generate the equal velocities. Lastly, the time of q's descent is equal to / x ^ 

 ^ap p_ j^gjj^^gg ^Yie times are always in the sub-duplicate ratio of the spaces 



directly, and forces inversely. 



It is now extremely easy to trace these expressions back again in a contrary 

 order, and to show, that if these last equations are true, the original one must 



be true also; that/>X v^ must necessarily be equal to — 7- -{• pv', or, which is 

 the same thing, that the body a multiplied into the square of its velocity, and 

 added to the sum of all the products which arise by multiplying every particle 

 into the square of its respective velocity, is equal to the body q multiplied by 

 the square of the velocity which it would have acquired by the same descent in 

 vacuo. 



Now this is to give the argument its full force; and since the conclusions are 

 confirmed by repeated experiments, as the author himself assures us, it is pre- 

 sumed, that the premises can be liable to no just exception. If we do not think, 

 with the advocates for this doctrine, that the vires vivae must always remain the 

 same from the thing itself, they will force our assent by the testimony of expe- 

 rience, and oblige us to admit their principles when we find it impossible to deny 

 the consequences. 



VOL. XIV. 3 C 



