different philosophers are frequently at variance; and, consequently, the theory cannot be said to have arrived at that state of perfection which is desirable.

233. The statical friction of plane surfaces is, under like circumstances, proportional to the pressure.

For let AB, ab be two planes in contact, placed in a horizontal position, the lower one AB being firmly fixed, but the upper one ab free to slide upon it. To ab attach a horizontal string bD passing over a pulley D, and having a dish C suspended from it. Load ab with a weight w, and denote the whole pressure of the plane ab on AB by W. Pour fine sand into the dish C until it begins to move, and then the weight of the dish and sand is the measure of the statical friction of the planes corresponding to the pressure W. If ab be loaded with more weights until the pressure is 2 W, the friction is found to be double of what it was before; when the pressure is 3W, the friction is trebled; and so on. Wherefore the statical friction of plane surfaces is proportional to the pressure.

This result was confirmed by Coulomb and Ximines for very considerable pressures; in extreme cases, where the pressures were very large indeed, the friction was observed to be rather less in proportion than for small pressures; the deviation from the above law was however so small, even for extreme cases, that we shall not fall into any very considerable error, in supposing the law to be universally true.

The following method of establishing the property of the proportionality of the friction to the pressure, is very convenient for experiments.

Let the body W (fig. 51) be placed upon an inclined plane AB, and then let the altitude BC be slowly increased until the plane has acquired such an elevation that W begins to slide down it; at this moment the friction just balances the weight W, and since it acts parallel to the plane in the direction AB, we may consider W as kept in equilibrium by a power in that direction, hence

cient of friction, and is taken as its measure. It appears then, that in the last experiment the coefficient of friction is

equal to the tangent of the inclination of the plane.

235. It being granted that the friction is proportional to the pressure when the surfaces are given, then, whatever be the magnitude of the surfaces in contact, the friction will remain the same, so long as the pressure is the same.

Let the body W (fig. 51) have faces, whose areas are C and D square inches; then when the first face is in contact with the plane, the whole pressure is supported on C square inches, and therefore the pressure on each square inch, is equal to

and therefore the friction upon each square inch of surface

In the same way it may be shewn that the friction upon the second surface

= (pressure). tan i,

and therefore the friction of a body is the same whether the surface on which it rests be large or small. When the surface is very small, the pressure on each square inch becomes very large, and then the friction, as observed in Art. 233, becomes somewhat less in proportion to the pressure; and therefore the friction is less, in a slight degree, when the body rests upon a small surface than a larger.

236. These are the chief properties of statical friction; it does not belong to us to investigate those of dynamical friction; but to make the subject complete we shall annex the following summary of results which have been obtained by various experimentalists.

(1) Dynamical friction is a uniformly retarding force: and it diminishes as the pressure increases.

This is only true when the surfaces in contact are hard; for in experiments made with bodies covered with cloth, woollen, &c. the friction was found to increase with the velocity.

(2) In the same body Statical friction is greater than Dynamical friction; i. e., it requires a greater force to put a body at rest in motion, than is requisite to preserve the motion undiminished when once it is produced.

This was thought by Professor Vince to arise from the cohesion of the body to the plane when it is at rest, which does not happen when the body is in motion.

(3) When a body of wood is first laid upon another, the Statical friction increases for a few minutes, when it attains its maximum, and no further alteration takes place. In making experiments, therefore, it is necessary to wait some time before the body is put in motion.

(4) Friction between substances of the same kind is greater than when they are of different kinds.

(5) The velocity has very little, if any, influence except when one body is composed of wood and the other of metal, in which case the resistance increases with the velocity.

(6) It is also found that friction is diminished by oiling and polishing the surfaces in contact. There is a limit however to the latter, for if they be very highly polished, the

resistance increases.

(7) The friction of cylinders rolling on planes, is proportional to their pressures directly and their radii inversely.

It is remarkable, that friction of this kind, unlike that between two planes, is not diminished by greasing or oiling the surface of the planes and cylinder. This kind of friction is much less than that produced by rubbing.

CHAPTER IX.

ON ELASTIC STRINGS.

237. STRINGS made of certain substances are found to be elastic; that is, they admit of being lengthened by the application of forces to their extremities, and regain their original dimensions, or nearly so, when the forces are removed. Spiral springs composed of steel wire, such as the one exhibited in fig. 71, are found to possess the same property in a remarkable degree. The connection between the force which stretches a string, or a spring of the kind here mentioned, and the increase of length cannot be investigated from mathematical considerations, but is to be determined entirely by experiments.

Let MN (fig. 72) be a very smooth horizontal table; AB an elastic string or spring laid upon it and fastened at A; W a weight stretching the string by means of a thread passing over the pulley C, whose position is such that ABC coincides with the table. Then, if W stretches the string to b, and another weight W' stretches it still farther to b', it is found that

Bb: Bb :: W : W';

that is, the excess of a given elastic string or spiral spring above its natural length is proportional to the weight which stretches it.

238. This excess is, in different strings of the same make and materials, proportional to their lengths.

For the tension of a string being the same in every part, if we divide the string into any number of equal parts, the increase of length in each part will be the same, and therefore