When a body moves in a discontinuous fluid in repose, the resistance it mects with is proportional to the square of the velocity; to the area of the section of the body moved, taken normally to the direction of the movement: it Is less In proportion to the length of that body in the direction of the movement ; and if two surfaces, covered one by the other, move in the same direction, the resistance of the covered face will be equal to a fraction of the face immediately exposed to the air; and the smaller the interval between the two faces, the leas will be the fraction in question. If, then, Q be called the resistance of the air sought, e = the area of the exposed end, v= the velocity of motion, E = a co-etticient depending upon the length of the train, and 0 = a constant co-efficient to represent the resistance caused by the intervals, the resistance of the air will be represented by the formula— Q=0 E A It thence follows that the sum of the calculable resistances to be overcome when a train moves in a straight line, on a level road, may be represented (calling the resistance r) by the formula, = fp r. f ( + p) + 0 € vt'.
When the train moves upon an incline forming with the horizon an angle a, the gravity of the train is decomposed into two portions, one of which is perpendicular to the plane of the rails, and the other acts in a direction parallel to that plane, and tends to draw the train down wards ; so that in pulling a train up an incline, the locomotive must exercise a power, not only able to overcome the friction, but also to overcome this effort of gravity in the direction parallel to the surface of the incline. The latter force, as is well known, is equal to (P+p) sin a ; the pressure of the bearings of the boxes upon the axles is equal tor cos a ; and the pressure of the wheels normally to the rails is equal to (P+p) cos a. The force, then, which the locomotive must exercise upon a waggon in order to allow the latter to retain the velocity it had at any particular moment, under the conditions supposed, must be fr cos a f+f(r +p) cos a +OE A v" (r + p) sin a. n As the inclines upon railways are generally such that we may consider cos a =1, and that sin a= t g a, we may consider that the force exer cised may, practically, be represented by the formula, frE..f'(r+p)+ The resistances a waggon encounters on a curve are of a complicated nature, and in a railway waggon they become more than usually so, from the fact that the wheels are fixed on the axles. Calling a the half width of the way, and p the mean radius of the curve, if the centre of the waggon travel over a space = 1, the interior wheels will travel and the exterior wheels so that the slip of each of them will be 1. Now the wheels exercise a pressure = r +p, and if we repre sent the co-efficient of friction by f", the expression of the resistance created by the fixity of the wheels will become f"(r + p) for every unity of distance. But there is another resistance developed on a curve from the two axles of a carriage being fixed to its frame ; for whilst the centre of the carriage traverses a circle described by the radius o (the mean radius of the curve), the respective points of con tact of the axles describe a circumference around the centre o of the rectangle formed by the points of contact of the wheels and of the rails. The radius corresponding to these points of contact is G A =
a= V, Is representing the half distance of the two axles, and a the half distance of the two points of contact of the wheels on the same axle. The slip of the wheels, whilst the waggon makes a revolution round o, becomes then f' (e p) 2 r + P. In the same space of time, the distance traversed by the waggon = 2 rp ; so that the resist. ;ince caused by the wheels being fixed on their axles, and by the parallelism of those axles themselves, is represented by the formula, 2r ‘,/ I " +p) (P — • There is a third force developed daring the curvilinear movement of a waggon arising from the pressure of the rims of the wheels upon the outer rail, whose expression is • ; in which g represents the accelerating force of gravity, which is in our latitude =32,1 feet. This r+p pressure gives rise to a friction, f'" in which f' represents the co-efficient of the friction created by the pressure of the rims of the wheels upon the inner surface of the outer rail. But in addition to this there is a friction exercised by the surface of the rim in its motion along the inner edge of the rails, and the resistance thus developed by the friction due to the centrifugal force, for every unity of the advance v' 2 R /1-1- of the waggon, is f" p ; in which n = the radius of the inner edge of the rim of the wheel, and h = the depth of that rim. The additional resistance thus created by the passage of a train over a curve is, then, represented by aia r +p V3ith f" (P + P) f p R The total effort which the locomotive must exercise in order to retain the velocity it previously had, must, during the unity of the distance traversed, be equal to the sum of the resistances above stated; and it is therefore represented by the formula Val+ b" T = fP + fr (P p) + (p + p) t g a + r + p) f/II P p V2 + g p rc In Perdounet's Traitd ElLimentaire; in Wood's ' Treatise on Rail ways,' in the second edition of 31. de Panabour's Tread des Machines Locomotives,' &c., the various experiments by which the values of the various coefficients in the above formulm have been ascertained, are discussed at length. It may suffice here to say that generally speaking the valises assigned by Coulomb to the coefficients f, f", are considered still to apply. though they are evidently in excess of the actual values as they have been indicated by experiments on railway trains; for Mr. Wood found that the total resistance of waggons of the old model travelling, at small velocities, upon a level straight line, was 0'00475 (p + p), and De Pambour found it to be only (r+p). Messrs. Gouin and Lechatellier found, by direct experiment with Morin's dynamometre, that the resistance from this cause was, at speeds vary ing between 15 to 25 miles per hour, equal to 0.003 to 0.0045 (P+p); at speeds between 25 and 38 miles, it was to 0.0085 (P +p); and for speeds between 50 and 60 miles, it was = to (r -Fp) On the average it may be taken that these resistances amount to (P +p).