736 PHILOSOPHICAL TRANSACTIONS. [aNNO IJSO. 



, sum of all the p X pC , ^ . , ^ r 



quently CD = ^^^^^7^7^^^,^^-;^^; therefore d is the centre of percussion, the 



point of suspension being at c. 



Cor. From the preceding prop, it appears, th;it every thing which was 

 proved in prop. 5, 6, 7, holds here also in the case of the action of one body on 

 another. 



Prop. 1 1. Let a body p (fig. 8) moving with the velocity v, strike the body q 

 at rest in the point A, and in a direction ad passing through the centre of gravity 

 of the striking body ; to determine the velocity of each body after the stroke, sup- 

 posing them to be elastic. — The solution of this prop, depending on the same 

 principles as that of prop. 3, we shall have, putting v equal the velocity of the 

 centre of gravity g after the stroke, on supposition that the bodies were non- 

 elastic (dgc being supposed perpendicular to ad, and c the point about which 

 the body a begins to revolve) v X p X cd = ULL^il^ + ^' x sum of all t].e p x cp^ 



CG CG 



and consequently v = ^ ,- ir l'' ^ ^"^f - , — ' j! but it is well known that 



' ■' sum 01 ail the p x pc^ + p x cd^ 



the sum of all the p X pc^ = cg X CD X a, and hence v = 



V ^ P ^ C G 



^-jTp ^c' ^"^ therefore if the bodies be supposed elastic, we have 



- — ^^ , ^ for the velocity of the centre of gravity g after the stroke. Now 



Q X CG -|- r X DC •' o j 



to determine the velocity of p, we have -- — ^ ^ ^° equal its velocity after the 



Q X CG + PX DC ^ -' 



stroke from single impact, and consequently v p x v x cd _ — «_x_v_x_cg — 



' •' Q X CG + P X DC Q X GC + P X DC 



is the velocity lost by p from simple impact; hence if the bodies be elastic, 



— will be the velocity lost by p if elastic, and consequently the 



velocity of p after the stroke = v - l2L^x_lJL£L = Li<_££_Z_12i^ x v 



Q X GC + P X DC Q X OC + P X DC 



Cor. 1. If the direction ad pass through g, then cg being equal to cd, we 

 have — ; — = a's velocity, and — ^ X v = p's velocity, which is well known 



Q + P ■' P + Q •' 



from the common principles of elastic bodies. 



Cor. 2. If p X DC = a X GC, or p : a :: gc : dc, then will the body p be at 

 rest after the stroke. 



Cor. 3. If a were infinitely great, the velocity of p after the stroke would 

 be = — V as it ought, for p would then strike against an immoveable obstacle. 



Cor. 4. Whatever motion a gains from the action of p, it would lose, if, 

 instead of supposing p to strike a, a were to move in an opposite direction, and 

 strike p at rest with the same velocity with which p struck a; in such case there- 

 fore, the velocity of a after the stroke would be 



2p X CrC X V (« — 2p) X gc + P X DC 



V = X V. 



Q X GC + P X DC Q X GC + P X DC 



Cor. 5. Hence, if p be infinitely great, or a be supposed to strike an 



