In order to find the 'pressure of a fluid against a body which is ter minated in front by a curve surface, an expression must be obtained (by means of the equation of the surface) for the area of an elemen tary portion of that surface, and this must be multiplied by the cube of the sine of Its inclination to the lino of motion. The product being multiplied by [ IlrottonasAmics ], and the whole inte grated between the proper limits, the result will express the required resistance.
Again, in investigating the motion of a body on an inclined plane when resisted by friction and the pressure of the atmosphere, the dal ' general equation of motion L--ag—a— may be employed. Here s tit is the space described in the time the velocity acquired In the same time, and dfa the differential expression for accelerative or retardative forte. If the body were to descend vertically, g, the force of gravity feet), would alone be the force producing the motion ; and the equation, being integrated, would give the relation — between the spaces described and the times of description when the body descends or ascends in a resisting medium. In the first of these eases g should be positive, and in the second negative. In order to adapt the equation to the descent of a body on an inclined plane, let B be the inclination of the plane to the horizon ; then y sin. 0 would represent the accelerative force on the plane if there were no friction. But since friction is proportional to the pressure (=g cos. 0) on the plane, and is independent of the velocity, let h be put for the coeffi cient of friction and represent a fractional part of the pressure ; thou we shall have hg coa 0 for the retardation produced by friction. a is the coefficient of the resistance due to the pressure of the atmosphere; it depends on the form and tnag,nitude of the moving body, and not on its weight ; and the resistance is suppoeed to be proportional to the square of the velocity, Thus the above equation becomes or, since the two first terms of the second member are constant, repro dal senting them by A, it becomes tits =A—a—. integrating this equa di' Om by successive approximations, or otherwise, we obtain in terms of the values ofeft (the velocity) and of s (the distance on the plane), either when the body sets out from a state of rest, or when it seta out with any given initial velocity. From these values, by means of the data obtained from good experiments, the values of h and a might be found ; and thus the effects of friction might be obtained separately from those which are due to the resistance of the air.
The forma° which are now generally received as expressing the resistances to which waggons moving upon railways are exposed are as follows :—They are of two classes • the one normal, and the other accidental ; the ono susceptible a priori calculation, the other depending upon the Mato of the road, the action of the wind, &c., or
upon conditions of an essentially variable nature. Of the normal causes of resistance, there are three kinds: 1, the friction of the axles hi the boxes : 2, the friction of the wheels upon the rails ; and 3, the resistance of the air, supposed to be in a state of repose, to the advance of the train. The level of the surface of the roadway naturally affects the numerical calculations of the various conditions thus specified ; but, as was before said (under RAILWAY), the introduction of the expansion gear into locomotive machinery has so modified the powers of that class of engine, that the importance of the precise value of the resistances to be overcome has of late been materially diminished. With an engine whose powers can be increased at will, and almost instantaneously, variable resistances are really matters of very little moment.. The formula for this class of resistance aro extracted from Perdonnct's Traiti Elimentaire des Chemins de Fer.' Taking into account, firstly, the case of a waggon moving in plain and on a dead level, it is evident that the horizontal movement must produce a friction of the bearing surface of the axle-boxes upon the axles themselves, which will be proportional to the pressure (or to the weight of the carriage, minus that of the wheels and axles), and will vary with the state of the bearing surfaces, but independently of their own state. If, then, the pressure upon the axle-boxes be represented by r, and the coefficient of friction by f (it will, in fact, be regulated by the nature of tho bearing surfaces, their (smoothness, and the nature of the lubricating material), the friction of the boxes on the axles will be represented by fr. Then, calling ft the radius of the wheels, and r the radius of the bearing of the axle, every revolution of the wheels will cause the waggon to advance a distance of 2 v n, and every point of the bearing a distance of 2 w r ; so that whilst the waggon advances through a distance = 1, the bearings of the axles will have 81iC1 over a The value of the friction of the bearings will then 2irn It The friction of the wheels against the rails is a friction of tho kind known as a rolling friction, and as such it is generally considered to be proportional to the pressure, and variable according to the nature of the surfaces in contact, but independent of the area of those surfaces, and of the speed of the motion. Or, calling p the weight of the wheels and axles, the total pressure or weight will be r +p, and f the coefficient of friction, the expression of this description of friction will then become f(r +p). Strictly speaking, the value of f' would depend on the size of the wheels ; but as, practically, the wheels of railway carriages are of the mute diameter, f' may be considered to be a constant quantity.