tion of motion remains constant under a wider range of pres sure than that of quiescence. It is moreover certain, that the limits of pressure beyond which the surfaces of contact begin to destroy one another or to abrade, are sooner reached when one of them is in motion upon the other, than when they are at rest: it is also certain that these limits are not independent of the velocity of the moving surface. The discussion of this subject, as it connects itself especially with the friction of motion, is of great importance; and it is to be regretted, that, with the means so munificently placed at his disposal by the French Government, M. Morin did not extend his experiments to higher pressures, and direct them more particularly to the circumstances of pressure and velocity under which a destruction of the rubbing surfaces first begins to show itself, and to the amount of the destruction of surface or wear of the material which corresponds to the same space traversed under different pressures and different velocities. Any accurate observer who should direct his attention to these subjects would greatly promote the interests of practical science.

SUMMARY OF THE LAWS OF FRICTION.

136. From what has here been stated it results, that if P represent the perpendicular or normal force by which one body is pressed upon the surface of another, F the friction of the two surfaces, or the force, which being applied parallel to their common surface of contact, would cause one of them to slip upon the surface of the other, and f the co-efficient of friction, then, in the case in which no unguent is interposed, f represents a constant quantity, and (Art. 133.)

F=ƒP. . . (109);

a relation which obtains accurately in respect to the friction of motion, and approximately in respect to the friction of quiescence.

137. The same relation obtains, moreover, in respect to unctuous surfaces when merely rubbed with the unguent, or where the presence of the unguent has no other influence than to increase the smoothness of the surfaces of contact without at all separating them from one another.

In unctuous surfaces partially lubricated, or between which

a stratum of unguent is partially interposed, the co-efficient of friction f is dependent for its amount upon the relation of the insistent pressure to the extent of the surface pressed, or upon the pressure per square inch of surface. This amount, corresponding to each pressure per square inch in respect to the different unguents used in machines, has not yet been made the subject of satisfactory experiments.

The amount of the resistance F opposed to the sliding of the surfaces upon one another is, moreover, as well in this case as in that of surfaces perfectly lubricated, influenced by the adhesiveness of the unguent, and is therefore dependent upon the extent of the adhering surface; so that, if S represent the number of square units in this surface, and a the adherence of each square unit, then aS represents the whole. adherence opposed to the sliding of the surfaces, and

P

where ƒ is a function of the pressure per square unit, and f a is an exceedingly small factor dependent on the viscosity of the unguent.

THE LIMITING ANGLE OF RESISTANCE.

We shall, for the present, suppose the parts of a solid body to cohere so firmly, as to be incapable of separation by the action of any force which may be impressed upon them. The limits within which this suposition is true will be discussed hereafter.

It is not to this resistance that our present inquiry has reference, but to that which results from the friction of the surface of bodies on one another, and especially to the direc tion of that resistance.

138. Any pressure applied to the surface of an immoveable solid body by the intervention of another body moveable upon it, will be sustained by the resistance of the surfaces of contact, whatever be its direction, provided only the angle which that direction makes with the perpendicular to the surfaces of contact do not exceed a certain angle called the LIMITING ANGLE OF RESISTANCE of those SURFACES,

This is true, however great the pressure may be. Also if the inclination of the pressure to the perpendicular exceed the limiting angle of resistance, then this pressure will not be sustained by the resistance of the surfaces of contact; and this is true, however small the pressure may be.

Let PQ represent the direction in which the surfaces of two bodies are pressed together at Q, and let QA be a perpendicular or normal to the surfaces of contact at that point, then will the pressure PQ be sustained by the resistance of the surfaces, however great it may be, provided its direction lie within a certain given angle AQB, called the limiting angle of resistance; and it will not be sustained, however small it may be, provided its direction lie without that angle. For let this pressure be represented by PQ, and let it be resolved into two others AQ and RQ, of which AQ is that by which it presses the surfaces together perpendicularly, and RQ that by which it tends to cause them to slide upon one another, if therefore the friction F produced by the first of these pressures exceed the second pressure RQ, then the one body will not be made to slip upon the other by this pressure PQ, however great it may be; but if the friction F, produced by the perpendicular pressure AQ, be less than the pressure RQ, then the one body will be made to slip upon the other, however small PQ may be. Let the pressure in the direction PQ be represented by P, and the angle AQP by, the perpendicular pressure in AQ is then represented by P cos. e, and therefore the friction of the surfaces of contact by fl cos. 4, f representing the co-efficient of friction (Art. 136.). Moreover, the resolved pressure in the direction RQ is represented by P sin. . The pressure P will therefore be sustained by the friction of the surfaces of contact or not, according as

P sin. is less or greater than fP cos. ; or, dividing both sides of this inequality by P cos. ê, ac cording as

tan. is less or greater than f

Let, now, the angle AQB equal that angle whose tangent is f, and let it be represented by p, so that tan. of. Substi tuting this value of f in the last inequality, it appears that the pressure P will be sustained by the friction of the sur faces of contact or not, according as

A

P

B

THE CONE OF RESISTANCE.

139. If the angle AQB be conceived to revolve about the axis AQ, so that BQ may generate the surface of a cone BQC, then this cone is called the coNE OF RESISTANCE: it is evident, that any pressure, however great, applied to the surfaces of contact at Q will be sustained by the resistance of the surfaces of contact, provided its direction be any where within the surface of this cone; and that it will not be sustained, however small it may be, if its direction lie any where without it.

THE TWO STATES BORDERING UPON MOTION.

140. If the direction of the pressure coincide with the surface of the cone, it will be sustained by the friction of the surfaces of contact, but the body to which it is applied will be upon the point of slipping upon the other. The state of the equilibrium of this body is then said to be that BORDERING UPON MOTION. If the pressure P admit of being applied in any direction about the point Q, there are evidently an infinity of such states of the equilibrium bordering upon motion, corresponding to all the possible positions of P on the surface of the cone.

If the pressure P admit of being applied only in the same plane, there are but two such states, corresponding to those directions of P, which coincide with the two intersections of this plane with the surface of the cone; these are called the superior and inferior states bordering upon motion. In the case in which the direction of P is limited to the plane AQB, BQ and CQ represent its directions corresponding to the

two states bordering on motion. Any direction of P within the angle BQC corresponds to a state of equilibrium; any direction, without this angle, to a state of motion.

141. Since, when the direction of the pressure P coincides with the surface of the cone of resistance, the equilibrium is in the state bordering upon motion; it follows, conversely, and for the same reasons, that this is the direction of the pressure sustained by the surfaces of contact of two bodies whenever the state of their equilibrium is that bordering upon motion. This being, moreover, the direction of the pressure of the one body upon the other is manifestly the direction of the resistance opposed by the second body to the pressure of the first at their surface of contact, for this single pressure and this single resistance are forces in equilibrium, and therefore equal and opposite. All that has been said above of the single pressure and the single resistance sustained by two surfaces of contact, is manifestly true of the resultant of any number of such pressures, and of the resultant of any number of such resistances. Thus then it follows, that when any number of pressures applied to a body moveable upon another which is fixed, are sustained by the resistance of the surfaces of contact of the two bodies, and are in the state of equilibrium bordering upon motion, then the direction of the resultant of these pressures coincides with the surface of the cone of resist ance, as does that also of the resultant of the resistances of the different points of the surfaces of contact*, that is, they are both inclined to the perpendicular to the surfaces of contact (at the point where they intersect it), at an angle equal to the limiting angle of resistance.

*The properties of the limiting angle of resistance and the cons of meist. ance, were first given by the author of this work in a paper published in the Cambridge Philosophical Transactions, vol. v.