VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 727 



the subject. Two years afterwards D. Bernoulli published a paper on progressive 

 and rotatory motion, containing nothing more than what I. Bernoulli had given 

 before, and, what is a little extraordinary, says in the introduction, de tali 

 quidem percussione nihil adhuc, quantum scio, publici juris factum fuit ab iis, 

 qui de motu corporum a percussione egerunt. Euler has also investigated the 

 velocities of the bodies after impact in a manner somewhat different, but has 

 rendered it much more intricate by a fluxional calculus. To any one however, 

 who attentively considers the subject, the theory must still appear to be extremely 

 imperfect, as, independent of principles not more self-evident than the proposi- 

 tions they are intended to demonstrate, which both I. and D, Bernoulli have as- 

 sumed in their investigations, a great variety of other circumstances, equally in- 

 teresting, naturally arise in an inquiry into this matter, and which are absolutely 

 necessary towards understanding the principles of the motion of the bodies after 

 impact. 



Prop. 1. Let a and b be tivo indejinitely small bodies connected by a lever void 

 of gravity ; and suppose a force to act at any point D, perpendicular to the lever ; 

 to find the point about which the bodies begin to revolve. — From the property of the 

 lever, the effect of the force acting at d, fig. 1, pi. 8, on the body a, is to the 

 effect on b, as bd : ad ; hence the ratio of the spaces aw, ^n, described by the 

 bodies a and b in the first instant of their motion, will be as — : — ; join mn, 



A B •* ' 



and if necessary produce that line and ab to meet in c, which will manifestly be 

 the point about which the bodies begin to revolve. Hence, from similar figures, 

 BC : AC :: — (oc b«) : — (oc atti) : : a X ad : b X bd, or DC — db : ad -f dc :: 

 A X AD : B X BD ; consequently do = a x ad + b x bd ^^^ therefore d is the 



^ ■' BXBD — AXAD 



centre of percussion or oscillation to the point of suspension c. 



Cor. 1 . Hence, whatever be the magnitude of the stroke at d, the point c 

 will remain the same. 



Cor. 2. If the force act at the centre of gravity g, the bodies will have no 

 circular motion ; for in this case b X bd — A X ad = O, and therefore dc be- 

 comes infinite. 



Cor. 3. If the force act at one of the bodies, the centre of rotation c will 

 coincide with the other body. 



Cor. 4. If the lever had been in motion before the stroke, the point c, at the 

 instant of the stroke, would not have been disturbed. 



Prop. 2. Let a given quantity of motion be communicated to the lever at d ; 

 to determine the velocity of the centre of gravity G. — ^The space Am, described 

 by the body a in the first instant of motion, is as — : now cg = cd — dg = 



+ A X AD^ + B X BD' , B X BD X BG + A X AD X AG 

 AD = AG 4- AD = — — ; 



B X BD — A X AD ' B X BD — A X AD 



