VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 737 



immoveable object, its velocity after the stroke will be = — -— X v: hence 

 when DC = "Zgc, the body q will have no progressive motion after the stroke, 

 but would in such case, if p were immediately taken away, continue to revolve 

 about a fixed axis. It may also be observed, that when dc is greater than 2gc, 

 or the velocity of a is positive, that, because it is impossible for a to continue 

 its progressive motion, it is only to be understood, that if immediately after the 

 impact the body p were removed, the body q would then proceed with such a 

 velocity. 



Cor. 6. Suppose the bodies to be non-elastic, and let m be the magnitude of 

 a body placed at d, which, being acted on by p, shall have the same velocity 

 generated as was before generated in the point d of the body a; then by 

 the common rule for non-elastic bodies, the velocity of m after the stroke will be 



PXV ,, PXV PXVXDC ., V, CG 



, and hence — • — = , consequently m = q x — . 



P+M P + M QXCG+PX DC ' •' DC 



Cor. 7- If a given quantity of motion were communicated to any point of 

 the body a, the progressive motion of that body after the stroke would be the 

 same. For suppose the magnitude of the body p to be diminished sine limite, 

 and its velocity to be increased in the same ratio, then ; because 



■' ' Q X CG 4- P X DC 



(which is the velocity of p after the stroke, if the bodies be non-elastic) = 

 (because p is infinitely small) '' ^ v x cd ^ ^^^ velocity of p after the stroke from 

 simple impact, is finite, consequently its motion must be infinitely small, and 

 therefore p must have communicated all its motion to a: now in this case the 



velocity of q ( = ^-^eT ^x^co ^ =~V"' "^^^^^ quantity is independent of the 

 place where the force acts; in the same manner it would appear if we had 

 supposed the bodies elastic. 



Prop. 12. Supposing evert/ thing given as in the last proposition, except that 

 the direction ad does not pass through the centre of gravity g of the striking 

 body; to determine the velocity of each body after the stroke. — Let ad (fig. 9) be 

 produced to meet f^o passing through g, the centre of gravity of the striking 

 body, perpendicularly in f, and suppose to be the point of the body p which 

 is not disturbed by the action of p on q : now it appears from cor. 6, prop. ] ] , 

 that if both bodies were non-elastic, and a body equal to q X — were placed at 

 D the velocity of that body, from the action of p, would be equal to the velo- 

 city of the point d of the body q ; for the same reason therefore it appears, that 

 if, instead of supposing p to strike a in the direction fa, a body equal to p x 

 — were to strike q at the same point, and in the same direction, which direction 



FO 



is supposed to pass through the centre of gravity of that body, the effect on o. 



V V 1* y c* r^ 



would be the same: hence, if in the quantity — , which from the 



VOL. XIV. 5 B 



