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34 THE CIVIL ENGINEER AND ARCHITECT’S JOURNAL. [January, the effective surface of the train to be 144 square feet, we must add 41 square feet per wagon, with the exception of the first, so that the effective surface will be found by adding to the area of the wagon of greatest section six square feet for the first, and 131 for each of the following wagons. Assuming the value of V, or the velocity at the foot of the first plane to he correctly given by the question in page 148, we found that the hypothesis of any thing approaching to uniformity of motion could not bv any means be reconciled with facts, but that by taking | V-' as the mean for the first plane, and V 2 for the second, the resistance of the air was correctly given by the equation we have quoted above. The square of the velocity at 1, $ and % of the length of the first plane are found by the above mentioned formula to be respectively equal to •326 V*, -G23 V ? and -864 V 2 . To simplify the calculation for general purposes a mean value of e, namely 1-05, which is suitable to a train of 15 wagons, is substituted in the above formula, which thus becomes, when the the velocity is expressed in miles per hour, Q =r -002687 2 v-. This chapter concludes with a practical table of the resistance of the air against trains at velocities commencing at 5 miles an hour, and increasing by 1 mile at a time up to 50, the effective surface of the train increasing by 10 square feet at a time from 20 to 100. The re sistance is expressed in lbs. on the whole train and on the square foot of effective surface. Chap. V. On the friction of the wagons on Railways. The only means of ascertaining the friction of wagons with any de gree of certainty is by the circumstances of their spontaneous descent and stop upon two consecutive inclined planes. We therefore pass to the 3rd section of this chapter, which is an analytical investigation of these circumstances, as referring to a system of two wheels joined to gether by an axle-tree fixed invariably to each, and loaded with a given weight resting on a chair on which the axle-tree may turn freely, “ The inquiry comprises three successive questions: 1st, To deter mine the effective accelerating force to which the centre of gravity of the system will be subject in its motion; 2nd. To deduce from this the velocity acquired by the moving body at the foot of the first plane; and 3rd. To conclude finally the distance it will have traversed on the second plane at the moment when the friction shall have reduced its velocity to nothing,” The motive forces applied to the system are first enumerated, in which the author includes, besides the motive forces properly so called, the passive resistances which oppose the motion, and which are gene rated by the motion itself. Among these there is one regarding which we think the author is in error, namely, the adhesion of the wheel on the rail. “ It is this force,” he says, “ which produces the rotation of the wheel, by preventing its circumference from sliding without turn ing during the motion along the plane.” This force is expressed by the weight T. If this ought to be looked upon as a force, there must also unques tionably be an expenditure of power without any resulting effect at the fulcrum of every lever, for, as the above quotation proves, it is only in its capacity of fulcrum that the point of contact of the circumference of the wheel with the rail is here considered; what is called the roll ing friction occupies the 6th and last place in the list. It is a curious fact that this introduction of a false idea does not in any way influence the final result of the calculation: it serves merely to form an unnecessary intermediate equation, between which, and the principal equation when the quantity T has been eliminated, the re sulting equation is the same as if that quantity had never entered into the calculation. The two equations in question are P sin 9' -f- p sin 6'—T —Q P <p, § and T R—/' Pr cos 9' —f" (P -\-p) cos 9* = - k‘ if, g in which P is the weight of the chair with its load, resting on the axle-tree, p that of the wheels aud axle-tree, 6' the inclination of the plane to the horizon, v the velocity of motion at any moment, Q r? the resistance of the air at that velocity, g the force of gravity, <p the effective accelerating force which produces the motion of translation of the system, if the effective accelerating force which produces the rotation of a point of the wheel situated at the distance 1 from the axle,^ k 2 the momentum inertias of the wheel, R the radius of the g wheel, r that of the axle, /' the coefficient of sliding friction, and/'' that of rolling friction. Now the former, or principal of the above equations ought evidently to have been P sin 9' -\-p sin S'-—/' P cos 0'-/” (P -L«) i cosff = K avir-zj, gR T p+/> ~T *• Substituting g for if, and 1 tor cos 9' as a sufficiently near approxi mation when the plane is but little inclined, and making r R ' " 1 ■ r ' R /' P | +/" (P +P) p =f (P +p), we obtain Whence (P +P) sin 6'-f (P -\-p) -- ^ _Q „2 _ L±H 0. g g <t>=~ 1 + 4-7F( sine '-/-PT7^ P +p R 2 This is precisely the equation arrived at by M. de Pambour, page 145, which is transformed, for the sake of simplicity, into the follow ing, <p—.g' (sin O'-f-q v-), the frictions represented by g' and q containing none but known quan tities. The accelerating force being equally represented by V d - (x being & X the distance traversed on the plane when the body has acquired the velocity v), this expression is substituted for <p, as well as h' for sin 9'—/, in the last equation, which thus becomes v d v v^# =gd *' which is the equation of the motion, and gives by integration between the limits x = o and x~l' = the length of the plane, calling V the velocity acquired at the end, whence sV ! = f (l- - 1 , ,Y H V Zqg'l') This gives the velocity at the end of the first plane, and conse quently at the beginning of the second. The question now is to de termine at what point of the second plane the body will stop, to solve which we have, calling 9" the inclination of this plane, all the other circumstances of the motion being the same as before, except that, the inclination of the plane being so much less, that the body is brought to rest, the accelerating force is negative, _ = -g' (b -f q b>), — b" being substituted for sin 6" — f. Making, after integration, X — l" for the distance traversed on the second plane, ;and v = o, since the body is brought to a state of rest, putting also for q V 2 its value found above, we have If b" - q g' i" -l ■qg' v-1 ‘<1 g' *’• Finally, restoring the values of g', b' and b"; and calling h' and h" the vertical heights descended on the first and second planes respec tively, and making