738 PHILOSOI'HICAL TRANSACTIONS. [aNNO 1780. 



last prop, expresses the velocity of the point d after the stroke, on supposition 

 that the bodies are non-elastic, we substitute for f a body equal to p x 



— , we shall have ; — for the velocity of the point d from 



the action of p ; and consequently x v x p x ^£211^ _ ^i^g velocity of 



^ •' Q X GC X lO -I- P X go X DC J "^l 



the centre of gravity g of the body q, after the stroke, if the bodies be perfectly 

 elastic. To determine now the velocity of the striking body, let of, perpendi- 

 cular to og, be the space described b\ ihe point o in the first instant of time 

 after the stroke, which, as that point is not disturbed by the action of the bodies 

 on each other, may represent the velocity of p before the stroke, and let f/; 

 represent the velocity of the point p after the stroke; join ft, and draw g-c/ 

 perpendicular to og, then will gd represent the velocity of the centre of gravity 

 o of the striking body after tlie stroke. Draw fc perpendicular to fa, and 

 produce gd to meet fc in e; now the velocity lost by p at the point p, by simple 

 impact, being equal to v — 



T X P'x DC X go V X Q X GC X FO 1 II 1 , , 



^-ir^^x-i^^^'go-x-^v - rx~,cx Fo+Pxgox'nc' ^^^ ^1^^" have be the ve- 

 locity lost by the point f, on supposition that the bodies are perfectly elastic 

 (supposing o'^to represent the vaUieofv) equal to — ?_xjLi<^21 "^ AZ^ gntl 



^ '^^ ° ' ' Q X GC X FO + P X go X DC 



therefore, by sim. triang. fc (fo) : cb v.fe iog) : ed = ^x^x^x^cxgo 



•' o ./ \ / J \ CI Q X GC X FO + P X go X DC 



= the velocity lost by the centre of gravity g, and hence v — 



2XVXQXGCXgO V X Q X GC X FO + V X P X go X DC— 2 X V X Q X CC X gO 



Q X GC X FO -I- P X go X DC Q X GC X FO + P X go X DC 



= the velocity of p after the stroke. Now as it appears, from prop. 9, that the 

 progressive motion of a body, when left to move freely, continues uniform and 

 in the same direction, it follows, that the expressions for the velocities of each 

 body in the first instant after the stroke, both in this and the preceding proposi- 

 tions, will represent the uniform progressive velocities with which the bodies will 

 continue to move, and consequently the place of each body, at the end of any 

 given time after impact, may easily be determined. 



Cor. \. If the direction fa pass through g, then, fo and go becoming infinite, 



, ,, , 2 X V X P X GC ,- , , . c J V X P X DC — V X Q X GC 



we shal have ^ — tor the velocity of a, and — ^ ~ 



Q X GC + P X DC •' Q X GC + P X DC 



for the velocity of p, agreeable to what was proved in the last proposition. 



Cor. 1. Hence the point about which p begins its rotatory motion, may easily 

 be found; for produce (if necessary)y& and of to meet in a, and a will be the 



2 X V X *4 X GC X FO 



point required; and by sim. triang k ( = -— ^-—-^--p-— -— _):c/::yb(=v) 

 . oa = ^ii_«^_L£J--L2i^^il£, and hence Fa = L.x g° .x "£:i^^ <>^ x °^ 



2 X Q X GC 2 X Q X GC 



Cor. 3, If, instead of supposing a to have been at rest, it had been moving 

 forward in a direction parallel to that of the body p, with the velocity i/^ the 



