- VOL. XLIII.] PHILOSOPHICAL TRANSACTIONS. 133 



^xiom 3. — A greater pressure produces a greater moving force, if the time 

 be given. 



Proj). 1. — Moving forces are not proportional to the masses of the bodies, 

 and the squares of their velocities. 



Demonslr. — Let there be two springs, equal, and equally bent, A and b, which, 

 by unbending themselves, push before them two unequal bodies, the spring A 

 pushing before it the greater body. 



Now, by axiom 1, the spring A will unbend more slowly than the other; from 

 which it follows, that at every instant of the time which the spring b takes up in 

 unbending itself, the spring A will have unbent itself less than b, or will be more 

 bent than b. Therefore, by axiom 2, the pressure of the spring a will, at any 

 instant of that time, be greater than the pressure of the spring b at that same 

 instant. Hence, by axiom 3, the nascent, or infinitely small moving force, 

 which is produced by the pressure of the spring a, in every infinitely small part 

 of that time, will be greater than that produced by the pressure of the spring b, 

 in the same infinitely small part of the time. 



Therefore, the sum of the infinitely small moving forces, that is to say, the 

 whole moving force, which is produced by the spring a, during that time, will 

 be greater than the moving force produced by the spring b in that same time: or 

 the moving force of the greater body will be greater than that of the less, at 

 the instant that the spring b, being now wholly unbent, ceases to act any longer 

 on the body it has pushed before it ; and as, after that instant, the spring a, not 

 being yet wholly unbent, continues to act on the greater body, the moving force 

 of the greater body will still continue to increase, and consequently will more 

 and more exceed the moving force of the smaller body. 

 ^ But every one knows, that the products of the masses and squares of the velo- 

 cities arc equal in the two bodies. Therefore the moving forces, which we have 

 proved to be unequal, are not proportional to the products of the masses and 

 squares of the velocities, a. e. d. 



To consider this in a particular example, let us suppose the masses of the two 

 bodies, exposed to the pressure of the springs a and b, to be 4 and 1 respec- 

 tively ; and let the spring b unbend itself, and thereby give the body 1 its whole 

 moving force in one second of time. Then, at the end of that second, the 

 moving force of the body 4 will already exceed that of the body 1, and will still 

 grow greater during another second of time. For the times are as the square 

 roots of the masses. Also, if the masses be 100 and 1, the moving force of 

 the body 100, will, at the end of the first second of time, be greater than 

 that of the body 1, and will continue to increase during the space of Q other 

 seconds. 



Carol. — When a bent spring, by unbending itself, drives a body before it, the 



