78 OF SIMPLE ACCELERATING FORCES. 



considered as a uniform force. Thus, if a body begins to 

 fall from a state of rest, it acquires in a single second a 

 velocity of 32^^ feet in a second ; and in two seconds a 

 velocity of 64^ feet : having described in the first second 

 16^ feet or 16-09, and in the second 32^+163^=48^. 

 The decrease of the force of gravitation, in proportion to 

 the square of the distances from the earth's centre, is 

 barely perceptible, at any heights within our reach, by the 

 nicest tests that we can employ. See 288. 



230. Theorem- The velocity, produced 

 by any uniformly accelerating force, is pro- 

 portional to the magnitude of the force, and 

 the time of its operation, conjointly. 



For, the time and the velocity both flowing equably, their 

 finite increments will be in a constant ratio (22j9, 47), and 

 the velocity being the measure of the force, the velocity 

 generated in a given time mast also be proportional to the 

 force. It may also readily be shown, by the composition 

 of motion, that a double action must produce a double velo- 

 city : for when the equal sides of a parallelogram, repre- 

 senting two separate motions, approach to each other, and 

 at last coincide in direction, the diagonal of the parallelo- 

 gram, representing their joint effect, becomes equal to the 

 sum of the sides: and the action of two independent 

 forces must be truly represented by the two sides of the 

 parallelogram, which represent them separately, otherwise 

 they would not be independent, nor could their combination 

 be called a double force. If we call the accelerating force 

 a, the time t, and the velocity produced v, we shall have v 



proportional to at, and — a constant quantity; or, if this 



quantity be called unity, at:=.v. 



