be loosed at the same instant from the top of a tall glass jar from the interior of which the air has been exhausted, they will both reach the bottom at the same time; or, in other words, the time of descent and the velocity acquired by each will be the same.

If a body be let fall from a height of, say, ten feet, the time of its descent being accurately taken; and it then be dropped from a height of twenty feet, or twice the height of the first experiment, the time occupied in its descent will be less than twice the time of the fall through ten feet. The motion is therefore accelerated.

Velocity. In the above example we have shown that the velocity acquired by a falling body is proportional to the time of descent. If the body in falling for one second acquires a velocity of, say, sixteen feet, the initial velocity of the second second of its fall is sixteen feet. But the force of gravity, being a constant force, will give an additional velocity of sixteen feet, as in the first second. The velocity at the end of the second second is therefore thirty-two feet.

The velocity generated by a constant force in any number of seconds, is the velocity gained in the first second multiplied by the number of seconds. If ƒ=the velocity generated in the first second, t the time, and v the velocity at the end of t, we have v=ft.

From this it is evident that the motion of falling bodies is an accelerated motion.

The motion of bodies falling in vacuo is uniformly accelerated; and the acceleration due to gravity is generally indicated by g. This acceleration is not the same for all parts of the earth's surface.2 In London 9= =32.2 feet, nearly, added in every second of time.

1 L., initialis, from the beginning. 2 Since the earth is not a perfect sphere and its centre of gravity does not correspond to its geometric centre, g therefore increases with the latitude of the place, for the surface of the earth is nearer to its C. G. at the poles than at the equator; and the force increases with propinquity to the C.G. For the same reason y decreases with the height above the sea.

Space. The velocity of the first second increases uniformly from 0 up to f; the mean velocity for that second is therefore f. The space passed over also=f. Again, the mean velocity for two seconds, is f, and the space passed over, 2f. Since f is passed over in the first second, f is passed over in the second second. In the same manner the spaces fallen through in the third, fourth, and fifth seconds are found to be f, f, and 1⁄2ƒ, respectively. The space described in t seconds will therefore be f (1+3+5+ ... + t). The spaces fallen through in each successive second are therefore as the numbers 1, 3, 5, 7, etc., which series has the property, that the sum of any number of terms is equal to the square of that number; thus the sum of the first two terms is 22, or 4; the sum of the first three terms is 3*, or 9, and so on.1

The spaces fallen through at the end of successive seconds, from the commencement of the motion are, therefore, as the numbers 1, 4, 9, 16, etc.; or, in other words, the space is proportional to the square of the time. Hence, to find the space fallen through from the commencement, we multiplyg (16 feet approximately) by the square of the number of seconds.

To find the space passed through in a particular second, we multiplyg (16 feet) by the odd number in the series 1, 3, 5, etc., which corresponds with the number of the second in the series 1, 2, 3, etc.

Thus, a body will fall 16×5=80 feet in the third second of its fall, the number 5 being third in the series of odd

multiply both sides of the second by 2 f, we have v2=f2 t2 and 2 fs =f2t2..v2=2fs. That is, the square of the velocity at any time is equal to twice the acceleration multiplied by the space already described. Wherefore, the acceleration of gravity being 9, v=gt; s=gt2; and v2 =2gs.

B

numbers, and at the end of that second the body will be 16×9-144 feet from its starting point, 9 being the square of the number of seconds.

The velocity at any point is found by multiplying g by the number of seconds from rest. In the above example the velocity at the end of the third second will be 96.6 feet nearly.

The height of a building or depth of a wall may therefore be found by multiplying the square of the number of seconds occupied by a stone in its descent by 16. Thus, if a stone dropped from the battlements of a tower reaches the ground in four seconds 42 = 16 and 16×16=256 ft. = height of tower.

Many watches are constructed to beat quarter seconds, and the intervals between each beat can easily be counted. If the square of the number of quarter seconds be taken, it will give the height. Thus, in the above example, the stone occupied 16 quarter seconds in its descent, and 162=256 ft.=the height.

Attwood's Machine.The law of falling bodies has been verified by means of an ingenious machine invented by Attwood, and called after him Attwood's Machine. It consists of an upright frame, A (fig. 109), having a scale attached. On the top of the frame is a pulley P, the pin of which turns on rollers to bring the resistance of friction to a minimum. Over this pulley a thread passes, bearing two equal and similar weights, B and C, which are therefore in equilibrium. G is a small weight capable of being

Fig. 109.

attached to B at pleasure. If it be placed on B the weight B will descend. The pressure = G, and the weight moved =B+C=2B. The acceleration is therefore

G

2B+G

and

this proportion can be made as minute as desired, by making the weight G very small. D is a clock constructed to beat seconds. Let B start from the division 1 of the scale, and let it fall for one second. Let 2 be its position at the end of the first second. If it be again let fall from 1 and stopped at the end of two seconds, it will have arrived at 3. Again let it fall from 1 for three seconds, and it will have reached 4. It will be found that the distance 1...3=4×1...2, the distance 1...4=9×1...2, and so on, showing the correctness of the rule, that the space is proportional to the square of the time.

Again, E is a ring capable of being moved up and down the bar A. Let it be so placed as to catch G as B passes 2, at the end of one second of time. B will then continue to move with the velocity it had when G was lifted from it. S is a stage also capable of being moved up and down the beam A. Let it be placed so as to catch B one second after it has passed E. ES will then represent the space described by a body moving with uniform velocity for one second. Again, let the stage S be placed at S2 so as to receive B two seconds after passing E. ES will represent the space moved over by a body in uniform motion for two seconds, and will be found to be double the space ES.

Hence, the velocity acquired is proportional to the time.

Graphic representation of acceleration of gravity. —If we take a length to represent seconds of time, as AB, BC, etc. (fig. 110), and mark off these lengths on a horizontal line, the length of AF will represent the time occupied by the body in its descent. From B, C, D, E, F, draw lines at right angles to A F, representing, according to the scale chosen, the spaces through which a body would fall in each consecutive second.

Thus, BG-16 feet; CH=64 feet; D I=144, and so on.. The distance through which the body falls in the

Fig. 110.

Fig. 111. first second being multiplied by the numbers 1, 4, 9, 16, gives the length of each consecutive line, B G, CH, etc. The points G, H, I, K, and L will represent the position of the body at the end of each successive second, and the curved line drawn through them will represent the acceleration of the velocity of the falling body.

Morin's Apparatus.-This curve has been marked out by an apparatus invented by General Morin.

It consists of a cylinder, A (fig. 111), caused to revolve about its axis by the descent of the weight B, attached to a cord wound round a horizontal axis, communicating by means of a toothed wheel with an endless screw on the axis of A. The uniformity of the motion is secured