14 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[January, 



the effective surface of the train to be 144 square feet, we must add 

 4\ 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 13i for each of the 

 following wagons. 



Assuming the value of V, or the velocity at the foot of the first 

 plane to be correctly given by the question in page 148, we found that 

 the hypothesis of any thing approaching to uniformity of motion could 

 not by any means be' reconciled with facts, but that by taking f V^ as 

 the mean for the first plane, and yV V" for the second, the resistance 

 of the air was correctly given by the equation we liave quoted above. 

 The square of the velocity at I, i and I of the length of the first plane 

 are found by the above mentioned formula to be respectively equal to 

 •32G V-, -623 V and •S64 V=. 



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 tlie the velocity is 

 expressed in miles per hour, 



Q = -0112(587 2 >•'. 



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 5U, the eflective 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 icagom 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 

 tire 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 wliieh oppose the motion, and which are gene- 

 rated by the motion itself. Among these there is one regarding which 

 we thirik 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, bv 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 rapacity 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 

 a)iy way influence the final result of the calculation: it serves merely 

 to form an unnecessary intermediate equation, between which, and the 

 princii)al equation when the quantity T has been eliminated, the re- 

 suiting equation is the same as if that quantity had never entered into 

 the calculation. 



The two equations ia question are 



P + o 

 Psine'-(-/)sine'-T-Q 1-== „.'''' 



and T R-/' Fr cos 6' -/" (P +p) cos fl' =^ k" +, 



in which P is the weight of the chair with its load, resting on the 

 axle-tree, ;; that of the wheels aud axle-tree, 8' 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' the momentum inertiae of the wheel, R the radius of the 

 g 



wheel, r that of the axle,/' the coefficient of sliding fi-ictioD, and/" 

 that of rolling friction. 



Now the former, or principal of the above equations ought evidently 

 to have been 



P sin fl' +p sin 6' -/' P ^ cos fl' -/ " (P+p)l, cos 9' -?- %. ,)/- Qtj-^ = 

 K K g K 



If). 



g 



Substituting p for if, and 1 for cos 8' as a suflBciently near approxi- 

 mation when the plane is but little inclined, and making 



/'P^+/"<P+P)^^=f(P+p), 



we obtain 



(P+p)sine'-f(P+p)--.~, ^_Qe^ = L±Z.^. 



Whence 



* = 



(sinfl'-/- 



1 + 



P+P 



■V')- 



P+p R- 



This is precisely the equation arrived at by M. de Pambour, page 

 145, which is transformed, for the sake of simplicity, into the foUow- 



?>—.§' (sine'-/-} tr), 



the frictions represented by g' and q containing none but known quan- 

 tities. 



V d V 

 The accelerating force being equally represented by — — (x being 



the distance traversed on the plane when the body has acquired the 

 velocity u), this expression is substituted for cp, as well as h' for sin 

 e'— /, in the last equation, which thus becomes 



„ = g'd.r, 



V d V 

 b'—q V? 



which is the equation of the motion, and gives by integration between 

 the limits ar = o and ;!: = Z'z=the length of the plane, calling V the 

 velocity acquired at the end, 



2qgl'= log. 



b' - q V 



whence 



9 V' = 6' (l- „ \ ,Y 



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 0" 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 



V d V 

 rest, the accelerating force , is negative. 



V d 



-^'(6"4-5c'), 

 — 6" being substituted for sin fl" — /. 



Making, after integration, x:=.l" for the distance traversed on the 

 second plane, ,and ;• = o, since the body is brought to a state of rest, 

 putting also for q V its value found above, we have 



'qg'l" 

 e -1 



'qg;l' 



Finally, restoring the values of g', b' and i"; and calling V and A" 

 tlie vertical heights descended on the first and second planes respec- 

 tively, and making 



