with reference to the Measure and Transfer of Force. 341 



them had been primitively in a state of repose : it is evident that 

 the space described by the body in consequence of its primitive 

 force, and of that which is added to it, becomes equal to the sum 

 of the spaces which each of them would have caused it to de- 

 scribe in the same time ; which supposes the force proportional 

 to the velocity." 



We may understand, from the instance of collision given in 

 § 10, that the fact of relative motion being the same whatever 

 the general motion of the system, is quite consistent with the 

 proportionality of force to square velocity. It may seem at first 

 somewhat of a paradox, that on striking a billiard ball in a west- 

 erly direction at noon so as to give it a velocity of 10 feet per 

 second, we should actually communicate to that ball such a force 

 or motion as would give it a velocity of 1500 feet per second if 

 the earth were at rest; but the difficulty disappears when we 

 consider that the force of impact or shock is made up of the differ- 

 ence between the sum of the absolute motions before impact, and at 

 impact or just before recoil begins, and that this difference is pro- 

 portional to the square of the relative velocity. The transference 

 of force from one body to another depends on the absolute velo^ 

 cities, so that it is physically possible to transfer any amount of 

 force, however large, by an impinging velocity, however small. 

 The shock of impact, and the amount of force transferred, do not 

 stand in the relation of cause and efi^ect. The one is not a func- 

 tion of the other, and the force of impact indicates nothing in 

 regard to the ''impressing of force'' or " ina'easing of force" 

 which depend on the absolute velocities, of which we cannot be 

 said to know anything certain. 



Take the instance of a ball swinging as a pendulum within a 

 railway carriage in the direction of its motion. Whether the 

 can'iage is at rest or in uniform motion at any speed, the motion 

 of the pendulum relative to the carriage is the same. This sup- 

 poses the mass of the carriage to be incomparably greater than 

 the ball. When they draw near to each other, as with a gunner's 

 eprouvette, the carriage sensibly reciprocates the oscillation of the 

 suspended cannon. Suppose between the pendulum ball and 

 point of suspension, a spring is interposed and the force on this 

 spring resolved in a horizontal direction. This horizontal part 

 accumulates to a maximum in one direction at the lowest part of 

 the arc of oscillation, then changes the direction of strain, and 

 accumulates to a maximum in the opposite direction ; in one-half 

 of the arc pulling forwards and in the other pulling backwards. 

 The amount of strain is like the foi'cc of impact, a function of 

 the relative velocity : the force transferred backwards and for- 

 wards from the carriage to the ball, and vice vei'sd, is a function 

 of the absolute velocities. 



