374 PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



testible, and which he himself" has admitted : for instance, he admits it as an 

 undoubted principle, that the quantity of motion in any system of bodies is pre- 

 served invariable, when estimated in a given direction, in all their collisions and 

 mutual actions on one another ; and in this he entirely agrees with the followers 

 of Sir Isaac Newton. Let us attend to the consequences of these two different 

 principles in the very case proposed by J. Bernoulli. And first, because ma = 

 mx-\- -^, by transposition we have m X {a — x) = -- , which is saying no 



more than that the motion lost by c is equal to the sum of the motions gained by 

 A and B, estimated in the same direction cd. By a similar process, from the 2d 

 equation, we deduce ?« X (« + 3:) X (a — x) = Inq'' ; and therefore the com- 

 parison of the two equations gives — z= y. The quantity y therefore, 



or the velocity of a or b after the stroke, must necessarily be equal to the sum 

 of the two quantities - and - . In the figure, let cd represent the velocity 

 of c before the stroke, and ch the velocity after it, and let fall the perpendicu- 

 lars wi, DL, on the direction AC. It easily appears, that en is equal to 

 -, and CL equal to^", because CH : cn :: CD : cl :: rad. : cos. lcd :: q : p. And 

 now the whole controversy is reduced into a narrow compass ; for whether the 

 two principles assumed by this author be consistent with experience or not, it is 

 impossible they should be consistent with each other, unless cn -j- cl shall be 

 found to measure the velocity of a in the direction cl. Suppose or to be the 

 velocity of c after impact, when all the bodies are perfectly hard, and letting fall 

 the perpendicular rs, cs will be the velocity acquired by a in that case ; and, 



universally, the velocity acquired by a will be equal to c.v -| — , if the elasticity 

 of the bodies be to perfect elasticity as i : m. In order to determine therefore 

 when cn -f cl can possibly be equal to cs -\- -, or, which is the same thing, 



LA' + cn equal to -, we are to consider that ns : L5 :: i : m : and because cn is 



equal to cs — sn, cn = c? — - , and it is obvious that C5 -I- L5 can never be 



equal to -, unless m be taken equal to unity, and Bernoulli's hypothesis is 

 plainly impossible in all cases where the bodies are not supposed perfectly 

 elastic. - 



But though we confess the learned author, who first solved the problem we 

 have been considering, deserves no commentlation for proposing in a general 

 form what ought to have been restrained to a particular case, yet it will by no 

 means follow, that every argument which has been advanced against this doctrine 

 is either intelligible or satisfactory. Of all the objections and experiments which 

 liave been started and contrived to refute the new opinions of the German phi- 



