to slide down the Plane; then in this case the proportion of the Friction to the Resistance is what we call the coefficient of friction for the state bordering on motion. Thus we have an experimental method of finding the value of this coefficient. The angle which the Inclined Plane makes with the horizon in the state of the body bordering on motion is called the angle of friction.

328. We have hitherto supposed that one body slides over another; the friction then may be called for distinction sliding friction. There is however another case in which the friction may be called rolling friction. Thus a solid cylinder may roll on a fixed plane, or within a fixed hollow cylinder; or a hollow cylinder may roll round a fixed cylindrical axis. It is found by experiment that in this case the friction is very nearly proportional to the pressure, but is much less than for the case of sliding surfaces kept in contact by the same pressure.

329. Friction may be diminished in various ways. Thus we may make the surfaces in contact very smooth; or we may interpose some lubricating material, such as oil or grease, between the surfaces in contact. It is found advantageous to have the bodies in contact of different substances; thus axles may be made of steel, and the parts on which they turn of gun metal or brass; in timepieces the steel axles often turn on agate or on diamond. We endeavour also as much as possible to avoid sliding friction and to introduce rolling friction; thus small wheels called castors are placed at the feet of tables and chairs for this purpose. Large masses of stone are often moved by the aid of many castors in the form of cannon balls placed under them. The following example has been given to illustrate the diminution of friction by various contrivances. A roughly hewn block of stone weighing 1080 pounds was drawn from the quarry on the surface of the rock by a force of 758 pounds. The stone was placed on a wooden sledge and then a force of 606 pounds was sufficient to draw it over a wooden floor. When the wooden surfaces in contact were smeared with tallow the force necessary to draw the stone was reduced to 182 pounds. Finally when the load was placed on wooden

rollers three feet in diameter the force was reduced to 28 pounds.

330. A very important contrivance is used for diminishing friction in the case of a body which turns round an axis in the way a grindstone does. The axis instead of resting on an immoveable support at each end rests on friction wheels as they are called. Two equal wheels are placed parallel and very near to each other; the distance between their centres is less than a diameter of the wheels. Thus at the upper part a kind of an angle is made on which rests one end of the axis of the body which is to turn. The other end rests on another similar pair of wheels. The friction wheels turn with the body, and the friction is found to be much less than it would be if the body turned on immoveable supports.

331. There are cases in which we find the assistance of friction very useful. Thus in frosty weather the iron rails become so slippery that the wheels of a locomotive engine turn round without biting the rails, and it is necessary to scatter a little sand on them to obtain the necessary roughness and consequent friction. When first railways were proposed it was maintained by some persons that the friction would always be inadequate to make the wheels bite, and that it would be necessary to cut teeth on the wheels and on the rails. Sometimes to procure enough friction we change rolling motion into sliding motion; thus the wheel of a carriage is locked when descending a hill in order to moderate the velocity by increasing the friction.

332. A remarkable case of friction is that which occurs when a rope is coiled round a solid body. Thus one end of a rope may be fastened to a barge, and if the rope is coiled two or three times round a strong post the barge will be easily held fast by a very small force at the other end of the rope, in spite of the current of the river in which it may be floating. The whole friction in this case increases very rapidly with the number of coils. Thus for example suppose that when the rope is coiled once round a force of one hundred weight supports eight hundred

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 Ỡ denote the point at which a horizontal axis passes through the