372 PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



ment which has yet been produced, where a clear judgment could he formed of 

 the effect, has confirmed the same thing. All the experiments usually brought 

 to determine the impressions made on soft bodies, as snow, clay, &c. are abso- 

 lutely unfit for the purpose. The circumstances, which take place in the pro- 

 duction of these effects, are such as we can never discover. The directions in 

 which the particles recede, the velocities they acquire, their mutual actions on 

 each other, and lastly the time in which these effects are performed, are all be- 

 yond tiie reach of computation. The other principle, that the relative velocity 

 of A and B is not altered by the stroke, is neither to be demonstrated nor con 

 firmed by experience ; it is a direct consequence of the definition of elasticity. 

 Again, suppose a. and )3 to represent the respective velocities of a and b after the 

 stroke ; then from these data it is easily inferred, that \yc -j- BfS'' = au^ -\- zb' : 

 for a — Z; is equal to j3 — a, because a — b is the relative velocity before, and 

 (3 — a. the relative velocity after the stroke. And xa -\- b/j is equal to ax -\- Bp, 

 because these quantities represent the sum of the motions before and after the 

 stroke respectively ; and from these equations the above equation is deduced ; 

 showing, that in elastic bodies the sum of the two bodies multiplied by the 

 squares of their absolute velocities, is not altered by the stroke. 



The same theorem may be demonstrated geometrically in the following manner. 

 Let the velocities of a and b be represented by ad, ab, respectively, fig. 7, 

 pi. 4 ; and let g be their centre of gravity, when placed at b and d ; the velocity 

 of A after the stroke will be represented by b^, if Gg be taken equal to gd, and 

 the velocity of b by ab -j- 2bg. From the nature of the centre of gravity, 

 A X GD = B X bg, and a X gd X 4ag = b X bg X 4ag = b X (4bg' -f 4bg 

 X ab). Add to both sides a X a^^ -j- b X ab", and we shall have a X ad* -f 

 B X ab^ = a X A^'- + b X (ab -|- 2bg^) 



We are not to wonder therefore, on making trials with perfectly elastic 

 bodies, if any such existed, were we always to find their vires vivae, as the 

 fcjreigners express themselves, neither increased nor diminished b) the stroke. 

 They define the force of bodies in motion, or their vis viva, to be in a compound 

 ratio of their quantities of matter, and the squares of their velocities ; and cer- 

 tainly such a definition implies no contradiction or impossibility. The term 

 force, in a loose and ordinary way of speaking, conveys to us no determinate 

 idea at all, and therefore, until it be defined, is incapable of being used to any 

 good purpose in philosophy : whether this or tiiat definition come nearer to the 

 general sense in which it is used indistinctly enough in common language, is 

 entirely another question. We may go farther, and add, that in their use of the 

 words, because the sum of the forces of elastic bodies is never affected by the 

 stroke, it is not unnatural to say, that action is therefore equal to re-action, and 

 that no force is lost by one body but what is communicated to the other. But 



