VOL. LXVIII.J I'HILOSOPHICAL TRANSACTIONS. 373 



if we will go so far, and thereby change the meaning of the terms action and 

 re-action and their measures, we ought at least to guard our readers from mis- 

 taking us, however convenient such modes of expression may appear. Because 

 A*- + B|3^ is equal to xa^ -\- Eb'^, it is true that no force is lost by a but what is 

 communicated to b ; but not in the same sense in which it was affirmed that no 

 motion is lost by a but what is communicated to b. In that case the squares of 

 their absolute velocities are understood ; in this, their velocities reduced to the 

 same direction. However, no material ill consequence can possibly arise from 

 such a notion of action and re-action, as long as the question is supposed to con- 

 cern only elastic bodies : but real mischief is done, and the debate ceases to be 

 verbal, whenever the law of the equality of action and re-action is said to take 

 place in the collisions of all sorts of bodies whatever. 



Case 2. — But the truth of these remarks, and the necessity of attending to the 

 precise use of terms, will appear in a still stronger light, if we consider the solu- 

 tion of a problem given by J. Bernoulli, in his Discours sur le mouvement. 

 Suppose that two equal and spherical bodies, a and b, fig. 8, struck at once in 

 the direction cd, perpendicular to the line joining the centres of a and b, with 

 a velocity represented by a. Let the quantity of matter in c be called m, and 

 the quantity of matter in a or b, n: let the velocity of c after the stroke be re- 

 presented by X, and that of a or b, in the direction ac or cb, by y, and sup- 

 pose/) : q :: rad. : cosin. lcd. iThen, because ma, the quantity of motion be- 

 fore the stroke, is equal to mx + ^—, the quantity of motion after the stroke, 

 and md^ is equal to nix^ + 2ni/', because the quantity of force is not altered by 

 the collision ; he easily finds x = '—■ — ;— rV-, and y = -j-'^;-— - , 



There is no problem which deserves to be more considered than this, by a 

 person desirous of having a clear idea of the grounds of that contention which 

 has subsisted so many years. We here see Bernoulli taking it for granted, that 

 the quantity of force in elastic bodies is no ways affected by their mutual actions, 

 whether direct or oblique ; and the most surprizing circumstance is, that he 

 should not so much as hint at any apparent difficulty in the present case, after 

 he had been so very diffuse in illustrating others which were much more simple. 

 No doubt he believed this principle to be a direct consequence of the equality of 

 action and re-action, and therefore it is plain he could not mean the same things 

 by those terms as we do at present. He believes no force is gained or lost by 

 impact ; he defines force by quantity of matter and square of the velocity con- 

 jointly ; and in estimating the velocity, he pays no regard to the direction in 

 which the bodies are moved. Let us not -cavil at his words : we cannot mistake 

 his meaning. The question is, how far these notions are agreeable to experi- 

 ence ; how far they are consistent with some other principles which are incon- 



