weight by the aid of the friction; then the same force will support 8 times 8 hundred weight when the rope is coiled twice round, and 8 times this when the rope is coiled thrice round, and so on.

333. Friction may naturally present itself to the reader at first in the light of an imperfection or obstacle in nature. By reason of friction the simplicity which we should otherwise often see in virtue of the First Law of Motion disappears. By reason of friction our machines never produce so much effect in moving bodies as they would otherwise. Nevertheless it is not difficult to shew that friction promotes in many respects the comfort of man, and a very interesting Chapter is devoted to the subject in Dr Whewell's Bridgewater Treatise; from this work the next two Articles are mainly derived.

334. The simple operations of standing and walking would scarcely be possible without the aid of friction; every person knows how difficult and how dangerous they are when performed on ice. Now there is really considerable friction in the case of ice, as we may see by the fact that a stone sliding on ice is brought to rest after it has gone but a slight distance. But the friction on ice is much less than on ordinary ground, and from our experience in moving on ice we may learn how embarrassing would be our condition on a perfectly smooth plane. At every step we take it is the friction of the ground which prevents the foot from sliding back, and thus allows us to push the other foot and the body forwards. And in the more violent motions of running and jumping it is easily seen that we depend entirely on friction for the possibility of the feat. Likewise when we wish to hold things in our hand it is friction which enables us to succeed; and on the contrary it was formerly the custom for wrestlers to rub their bodies with oil that they might be less easily grasped by their adversaries. Again the objects which surround us in our rooms, as chairs, tables, and books, would yield to the slightest push or current of air, and be in a state of perpetual motion if it were not for friction. The stability of our buildings is largely due to friction. It is true that mortar is used to assist in binding the bricks

and stones together, but were it not for friction the strength of the mortar would be always on trial as it were, at every shock and every breeze; and would give way under the long continued strain. But owing to friction the stability would subsist in many cases even without the mortar, and thus the tenacity of the mortar is reserved as it were for extreme occasions.

Were it not for friction rivers that now flow gently would be converted into rapid torrents. By the aid of friction we can form long threads and sheets out of the short fibres of cotton, flax or hemp; for it is friction consequent upon the mutual pressure of the fibres which are twisted together that keeps the material of these fibres together.

335. It is remarkable that friction which is so important in the concerns of the world disappears almost entirely when we turn to the larger motions of the heavenly bodies. All motions on the earth soon stop, but the moon and the planets continue in their courses for ages. So great is the apparent difference that the ancients were quite misled, and divided motion into two kinds, natural like that of the heavenly bodies, continually preserved, and violent like that of earthly objects, soon extinguished. Modern philosophers maintain that the nature of motion is the same, and the laws the same, for celestial and terrestrial bodies; that all motions are natural, but that in terrestrial motions friction comes into play and alters their character. Moreover there is strong reason for believing that all space is occupied by a medium, which though excessively rare does impede the motions of the heavenly bodies.

XXIV. GENERAL MOTION.

336. WE have more than once drawn attention to the circumstance that the motion with which we have been concerned is of a simple and restricted kind. We have spoken of it as the motion of a particle, and as the motion of a body where all the points move in the same manner, and as excluding all motion of rotation; see Arts. 123 and 285. The motion of bodies considered without this

restriction is beyond an elementary work like the present, and we must confine ourselves to a very few remarks respecting it. One of the most simple cases is that of motion round a fixed axis. Take, for example, the diagram of Art. 220, and suppose that P and W are not in the proportion necessary for equilibrium. Then motion ensues; one of the two, P and W, descends and the other ascends, while the piece consisting of the Wheel and Axle turns round a fixed horizontal axis. Suppose that W is larger than it ought to be for equilibrium; then W descends, and it is found by theory that W moves down with a velocity which increases in the same proportion as the time, that is W moves in the same manner as a body falling freely; but the motion is less rapid than that of a free body. Instead of the number 32 of Art. 92 we have now a smaller number, the value of which depends on P and W and on the weight and size of the machine. Also P ascends according to the same law, but with another number instead of 32. If the machine is very small and light compared with P and W its own motion will be unimportant, and we have very nearly the same case as that in Art. 142.

337. In the preceding case we have a body which can turn round a fixed axis, and which is kept in motion by the action of constant forces, namely P and W. But such motion might be produced by the action of forces which are not constant. For example, in raising water from a well the hand which turns the machine might exert force irregularly, sometimes more and sometimes less; and then the ascending body would no longer move like a body under the influence of gravity only.

338. We have spoken in Art. 312 of a simple pendulum, and have defined it as a heavy particle at one end of a fine string, the other end being fixed. But this is rather an ideal pendulum than a really existing object. A real pendulum may be defined to be a body of any form which can turn round a fixed horizontal axis.

Let AB be a body of any form, as for instance a rod with two fixed balls, one near each end. Suppose the plane of the paper to be vertical, and let denote the point at which a horizontal axis passes through the

A

body. The ends of this axis are supported; and the body being drawn away from its position of equilibrium and then left free will move to and fro. Now it is found by theory that the motion of this real pendulum is exactly the same as that of a simple pendulum of some definite length which can be calculated when the form and the substance of the body are known; this length is called the length of the equivalent simple pendulum. If we measure along the line through C and the centre of gravity of the body, a distance CO equal in length to the length of the equivalent simple pendulum, then O is called the centre of oscillation, while C is called the centre of suspension, The centre of gravity of the body will be at some point between C and O.

B

339. It is remarkable that the centres of oscillation and suspension are convertible; this means that if the body instead of turning round the horizontal axis at C turns round a parallel axis at O, then C becomes the new centre of oscillation.

340. The position of the centre of oscillation can be determined as we have said by theory; but it may also be found by experiment. For example if a slender rod oscillate about an axis through one end at right angles to the rod it is found that it oscillates in the same time as a simple pendulum two thirds of the length of the rod. Thus the centre of oscillation is distant two thirds of the length of the rod from the fixed rod. The statement can be verified by making the rod oscillate about an axis through the point thus assigned; then by Art. 339 the time of oscillation will be the same as before.

341. The rule found by theory for the length of the equivalent simple pendulum in the case of any body is the following. Suppose the body to consist of any number of

equal small particles, then the required length is a fraction to be calculated thus: The numerator is the sum of the squares of the distances of the particles from the horizontal axis; the denominator is the sum of the distances of the particles below the horizontal plane through the axis from that plane, diminished by the sum of the distances of those above, when the body is in its lowest position.

342. The centre of oscillation does not necessarily fall within the body. It is obvious from the diagram of Art. 338 that the parts of the body above and below Crespectively are always tending by their weights to move the pendulum in contrary directions, so that if these two parts are so adjusted as to produce nearly equal effects the motion may be very slow indeed, and thus CO may be very long and consequently the point O quite beyond the body. Musicians use a small pendulum called a metronome for the sake of marking time; though very short it can be made to oscillate in a second or even in a longer time. It is of the form represented in the diagram, namely a rod with balls at the ends. The upper ball can be moved to any position which may be desired, and held fixed in that position by a screw; thus the metronome can be made to Oscillate at a quicker or slower rate as may be required.

343. It will be observed that a heavy body oscillates in the same manner as if the whole weight were collected at the centre of oscillation, not as if it were collected at the centre of gravity. It is sometimes stated incautiously that the weight of a body may always be supposed to be collected at the centre of gravity; but the present case shews that such a statement is too wide: see Art. 169.

344. The following very important result is demonstrated by theory with respect to the motion of any body. The motion of the centre of gravity of a body is exactly the same as the motion of a particle having a mass equal to the mass of the body and acted on by forces equal and parallel to those which act on the body. The reader will scarcely be prepared to understand completely this very remarkable statement, but even an imperfect notion of it will be of service. Take, as a simple example, a top spin