obtain, in respect to that friction which accompanies motion, with a precision and uniformity never before assigned to them; they have given to all our calculations in respect to the theory of machines (whose moving surfaces have attained their proper bearings and been worn to their natural polish) a new and unlooked-for certainty, and may probably be ranked amongst the most accurate and valuable of the constants of practical science.

It is, however, to be observed, that all these experiments were made under comparatively small insistent pressures as compared with the extent of the surface pressed (pressures not exceeding from one to two kilogrammes per square centimeter, or from about 14-3 to 28.6 lbs. per square inch). In adopting the results of M. Morin, it is of importance to bear this fact in mind, because the experiments of Coulomb, and particularly the excellent experiments of Mr. G. Rennie, carried far beyond these limits of insistent pressure*, have fully shown the co-efficient of the friction of quiescence to increase rapidly, from some limit attained long before the surfaces abrade. In respect to some surfaces, as, for instance, wrought iron upon wrought iron, the co-efficient nearly tripled itself as the pressure advanced to the limits of abrasion. It is greatly to be regretted that no experiments have yet been directed to a determination of the precise limit about which this change in the value of the co-efficient begins to take place. It appears, indeed, in the experiments of Mr. Rennie in respect to some of the soft metals, as, for instance, tin upon tin, and tin upon cast iron; but in respect to the harder metals, his experiments passing at once from a pressure of 32 lbs. per square inch to a pressure of 1.66 cwt. per square inch, and the co-efficient (in the case of wrought iron for instance) from about 148 to 25, the limit which we seek is lost in the intervening chasm. The experiments of Mr. Rennie have reference, however, only to the friction of quiescence. It seems probable that the co-efficient of the fric

* Mr. Rennie's experiments were carried, in some cases, to from 5 cwt. to 7 cwt. per square inch.

tion of motion remains constant under a wider range of pressure 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

F=ƒP+aS. . . . . (110);

P

where ƒ is a function of the pressure per square unit and

α

S'

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 supposition 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 direction 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 0, the perpendicular pressure in AQ is then represented by P cos. 0, and therefore the friction of the surfaces of contact by fP cos. 0, 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 ƒP cos. ;

or, dividing both sides of this inequality by P. cos. 9, according 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 4, so that tan. =f. Substituting this value of ƒ in the last inequality, it appears that the pressure P will be sustained by the friction of the surfaces of contact or not, according as

tan. is less or greater than tan. 7,

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 sur

faces 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