VOL. LXVIII.] PHILOSOPHICAL TRANSACTIOxVS. 287 



of the point whose distance is k, we have this proportion, as Z- : h :: z : 

 "-7-=^ 77 = the velocity of this compound centre of oscillation. 



Again, since — is the versed sine of the described arc c, its radius being /•; 



therefore ?: h ::—:-— X -r , , = the versed sine of the radius h, or the 



versed sine of the arc described by the centre of oscillation, which call v; then 

 is V the perpendicular height descended by this centre, and the velocity it ac- 

 quires by the descent through this space is thus easily found, viz. as y/iO-J^ : 

 ■v^ v :: 32-L : - — - X t/ , , '^ = the velocity of the centre of oscillation deduced 



" r^2 bk + gp ■' 



from the chord of the arc which is actually described. 



Having thus obtained 2 difterent expressions for the velocity of this centre, 

 independent of each other, let an equation be made of them, and it will ex- 

 press the relation of the several quantities in the question ; thus then we have 



bkv 8.02c ,bkk + ghp r i- . i. ■ 8.02c ,r,7i , \ 



,— - — = ■ — -■ a/ .,, , from which we obtam v = rL~7^ ^ ('''■ + f;P) X 



hk + gp r^2 bk-\-gp bkr ^2 LV > t'l I ■^ 



{bkk -\- ghp)\ the true expression for the original velocity of the ball the moment 

 before it struck the pendulum. 



CoKOL. But this theorem may be reduced to a form much more simple and fit 

 for use, and yet be sufficiently near the truth. Thus, let the root of the com- 

 pound factor \/ \_{bh -{• gp) X {hhk -\- ghp)'] be extracted, and it will be equal to 



V h X [pg + L'k X -4j;-) within the 100000th part of the truth in such cases 



h + k 



as generally happen. But since bk x — r- is usually but about the 300th or 



h + k 



400th part of pg, and that bh differs from bk X -^ but by about the 80th or 

 1 00th part of itself, therefore pg -f- bk is within about the 20000th or 30000th 

 part o(pg -f bk X -^. Consequently v is = 8.02c V ^h X ^'^^ very nearly. 

 Or, further, if g be written for k in the last term bk, then finally v is := 8.02 

 eg s/ jLh X ^-rj—> or f = 5.672cg- \//j X ^ttt-; which is an easy theorem to be 

 used on all occasions; and being within about the 3000th part of the truth, it is 

 sufficiently exact for all practical purposes whatever. Where it must be observed, 

 that c, g, k, r, may be taken in any measures, either feet or inches, &c. provided 

 they be but all of the same kind; but h must be in feet, because the theorem is 

 adapted to feet. 



Scholium. As the balls remain in the pendulum during the time of making 

 one whole set of experiments, by the addition of their weight to it, both its 

 weight and the centres of gravity and oscillation will be changed by the addition 

 of each ball which is lodged in the wood, and therefore p, g, and //, must be 

 corrected after every shot in the theorem for determining the velocity v. Now the 

 succeeding value of p is always p -]- b; or p must be corrected by the continual 



