30 Mr. HerapatlCs Reply to X, [Jan. 



sums of the collisions in a given or an indefinite time ; that is, by 

 the product of a single collision and the number of them, or by 

 the product of a single momentum, the number of acting parti- 

 cles, and number of returns. Had I, therefore, omitted the fac- 

 tor X. objects to, I should have committed a sad error ; it would 

 have been like endeavouring to equate a single impulse with an 

 unceasing force for an indefinite time, — a manifest impossibihty. 

 To illustrate this, the best course is, perhaps, that which I 

 have already pursued, p. 341 ^////cr/s for May. Let the perfectly 

 hard ball A be continually solicited in the vertical direction C D 

 by some uniform force, such as that of gravity ; and when it ha^^ 

 descended to E, and acquired the velocity a, let it be met by 

 another perfectly hard ball B, not impelled by this gravitating 

 force, having a contrary velocity Z>; so that B 6 = A «. Then 

 the opposing momenta being equal by Prop. 5, of my 

 theory of collision, A will begin to reascend with an 

 equal momentum A a ; and being still acted on by by the 

 invariable gravitating force, it will continue to ascend until |^ ^ 

 all its motion be destroyed. After this, it will again 

 begin to descend, and at E will have the same momentum 

 as before. If now it be a second time met by the ball B 

 with the momentum B 6 = A g, it will a second time re- 

 ascend and descend in precisely the same way as at the ~ E 

 first. The circumstances of a third, fourth, &c. collision 

 being the same, the phsenomena of a third, fourth. Sec. 

 ascent and descent will be the same ; and thus the effect 

 of gravity on the one will be counteracted by the equal 

 and uniform colHsions of the other. Let /' be the force ■" 

 of pressure or gravity, t the time of acquiring the motion 



A a ; then 2/ 1, 4/ 1, 6ft 2 7ift are the effects of D 



gravity to be overcome by 1, 2, 3, . . . > n collisions ; so 

 that after the nth collision, or after a certain time T, the effect 

 of gravity overcome is 2 nf t — n A a = n B a. If, therefore, 

 T be accounted from the commencement of the descent of A to 



the completion of the nth contact, we shall have -- — = (w~l), 



and/ = 1. Hence tiB a = ^ , - But if t be taken 



indefinitely small, the oscillations of A will not sensibly change 

 it from a state of apparent rest, and in that case n for any given 

 time T must become exceedingly great, so that n B a =J'T, 

 Putting, therefore, T = 1, we have y* = n B a; that is, the 

 force of pressure, or, which is the same, the action of the ball 

 to support that pressure, is equal to a single momentum of the 

 ball multiplied by the number of returns in a unity of time. Thus 

 X. will perceive that the subject admits of a rigid mathematical 

 proof. 



X. asserts that I have, ''by my own confession, assumed an 

 hypothesis producing a result at variance with experiment." 

 Surely X. must have been curiously mistaken. He cannot 



