286 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1778. 



required, till the 1 ends exactly balanced each other; then, as it lay, the dis- 

 tance was measured from the middle of the axis to the part which rested on the 

 edge of the prism, or the centre of gravity of the pendulum. 

 The other method was as in the latter of the 1 annexed 

 figures, where the ends of the axis being supported on fixed 

 uprights, and a chord fastened to the lower end of the pen- 

 dulum, was passed over a pulley at p, different weights w 

 were fastened to the other end of it, till the pendulum was 

 brought to a horizontal position. Then, taking also the 

 whole weight of the pendulum, and its length from the 

 axis to the bottom where the chord was fixed, the place of 

 the centre of gravity is found by this proportion, as p the 

 weight of the pendulum: w the appended weight :: d the 

 whole length from the axis to the bottom : — =i the distance from the axis to 

 the centre of gravity. Either of these 1 methods gave the place of the centre 

 of gravity sufficiently exact; but the coincidence of the results of both of them 

 was still more satisfactory. 



Of ike Rule for Computing the Felocitij of the Ball. — Having described the 

 methods of obtaining the necessary dimensions, we proceed now to the investi- 

 gation of the theorem by which the velocity of the ball is to be computed. The 

 several weights and measures being found, let then b denote the weight of the 

 ball, p the whole weight of the pendulum, g the distance of the centre of gra- 

 vity below the axis, h the distance of the centre of oscillation, k the distance to 

 the point struck by the ball, z the velocity of this point struck after the blow, v 

 the original velocity of the ball, c the chord of the arch measured by the tape, 

 and /• its radius, or the distance from the axis to the bottom of the pendulum. 



ah 



Then the effect of the blow struck by the ball is as ^p or kk : gh :: p 



^ = the 



weight of a body, which, being placed at the point struck, would acquire the 

 same velocity from the blow, as the pendulum does at the same point. Here 

 then are 2 bodies, b and ^rr, the former of which, with the velocity v, strikes 

 the latter at rest, so that after the blow they both proceed uniformly forward 

 together with the velocity z; in which case it is well known that b : b -\- 



S'>P .. 



z: V : 



and therefore the velocity z is = ,y 



bkkv 



But because of the ac- 



kk " " "■" •' ~ ■" " bkk + ghp' 



cession of the ball to the pendulum, the place of the centre of oscillation will be 



changed; and from the known property of that point we find -77 — f-^ = to its 



Call this distance of the centre of oscillation, of the 



distance from the axis. 



mass compounded of the ball and pendulum, h. 



Then, since z is the velocity 



