MECHANICS. 



15 



had acquired at R. Now we have seen 

 that this velocity was such that it moved 

 through six inches in one second. 



E.t: 2. Again, let the stage be placed 

 so that when the weight A rests upon 

 it, the top of the weight will be twelve 

 inches from the ring R, and let the 

 weight B be depressed until the top of 

 the weight A is twelve inches above the 

 ring R.~ This done, let the bar F be 

 placed on the weight A, and let that 

 weight be disengaged at the moment of 

 any beat of the clock ; it will be ob- 

 served that the stroke of the bar F upon 

 the ring R will coincide exactly with the 

 third beat, the descent through twelve 

 inches being made in two seconds, and 

 that the stroke of the weight A upon the 

 stage S will coincide precisely with the 

 fourth beat, the weight moving through 

 the twelve inches below the ring with 

 the velocity it has acquired in two 

 seconds. 



Ex. 3. Now let the stage S be once 

 more removed, and placed so that, when 

 the weight A stands upon it, the top of 

 the weight will be eighteen inches below 

 the ring R. Let the weight B be de- 

 pressed until the top of the weight A is 

 twenty-seven inches above the stage S. 

 Let the bar F be then placed upon the 

 weight A as before, and permitting the 

 weight to commence its descent with the 

 first beat of the pendulum, the bar will 

 strike the rins: R with the fourth beat, 

 and the weight A will strike the stage S 

 with the fifth beat. The weight, there- 

 fore, descends through twenty- seven 

 inches with an accelerated motion in 

 three seconds, and at the end of that 

 time has acquired such a velocity, as to 

 move through eighteen inches in a 

 second. 



(39.) Now let us review the results of 

 these three experiments. By the first it 

 appears, that the velocity acquired in 

 one second is such as to make the weight 

 A move at the rate of six inches per 

 second. By the second experiment it 

 appears, that the velocity acquired in 

 two seconds is twelve inches per second ; 

 and by the third experiment it appears, 

 that the velocity acquired in three 

 seconds is eighteen inches per second. 

 Thus the velocities acquired in one, two, 

 and three seconds, are as six, twelve, 

 and eighteen, which numbers are as one, 

 two, and three. Hence the law before 

 explained, that" the velocities acquired 

 are as the time of acquiring them," is 

 verified. 



In the first experiment the weight fell 



through three inches in one second ; in 

 the second experiment it fell through 

 twelve inches in two seconds, and in the 

 third it fell through twenty- seven inches 

 in three seconds^ Now the numbers 

 three, twelve, and twenty-seven are as 

 one, four and nine, whichare the squares 

 of one, two, and three. Hence the law 

 already explained, that " the spaces fallen 

 through are proportional to the squares 

 of the times," is verified. 



In the first experiment it was shown 

 that the velocity acquired in falling 

 through three inches, was such as would 

 carry the weight in the same time 

 through six inches when continued 

 uniform and without further increase. 

 In the second experiment it was shown, 

 that with the velocity acquired in falling 

 through twelve inches in two seconds, 

 the weight A would move through 

 twelve inches in one second, and it 

 would, therefore, move through twenty- 

 four inches in two seconds. In like 

 manner, in the third experiment, it ap- 

 peared that with the velocity acquired 

 in falling through twenty- seven inches 

 in three seconds, the weight A moved 

 through eighteen inches in one second ; 

 and, therefore, would move through 

 fifty-four inches in three seconds. Each 

 of 'these experiments, therefore, veri- 

 fies the law, that, " with the velocity 

 which a body acquires in any time, it 

 would, if that velocity were continued 

 uniform, move through twice that space 

 in the same time." 



Also by the first experiment it ap- 

 peared that the space fallen through in 

 the first second of the descent was three 

 inches. By the second experiment it 

 appeared, that the space fallen through 

 in the first two seconds was twelve 

 inches. It consequently follows, that 

 the space fallen through in the second 

 second must have been nine inches. 

 By the third experiment the space fallen 

 through in three seconds was twenty- 

 seven inches. Taking from this the 

 space fallen through in the first two 

 seconds, which is twelve inches, the re- 

 mainder, fifteen inches, is the space fallen 

 through in the third second. Thus the 

 spaces described in the first, second, 

 and third seconds of the fall are three, 

 nine, and fifteen inches respectively, 

 which are as the numbers one, three, 

 and five. This verifies the law before 

 explained, that " the spaces described 

 by a falling body in the successive equal 

 intervals are as the odd integers." 



Since the heights from which bodies 



