384 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[October, 



The inimciliate consequence, from the friction l)eing the same at all velo- 

 cities, is, tliat it is a uniformly relanUng force, that is to say, that it destroys 

 in the moving body on w liieh it acts equal velocities in equal tunes. This, 

 if it destroy a certain amount of sjiced at the cud of one second, it \\U] 

 destroy twice that at the end of two seconds, three times at the end of three 

 seconds, and so on. Now if a railway train, moving down a steep inclined 

 plane, were suljject to no other resistances than those arising from friction, 

 it is evident that it would move down the plane with a uniformly accelerated 

 motion, although that motion would he less accelerated than if it were sub- 

 ject to no resistance. In other words, the kind of motion affei ting it would 

 be the same as if tliere was no resistance, the rlri/rce of motion alone beitig 

 altered. It has been stated that, subject to no resistance, certain speeds 

 would be gained by the train in one, two, three seconds. These speeds 

 would be those due to the gravity of the plane. These speeds would, how- 

 ever, now be diminished by the amount of velocity destroyed 1)y the fiiction ; 

 and as this latter would lie increased in the same proportion as the sjiecil 

 imparted by gravity, the descending body woulil be accelerated by a uuifoim 

 force, equal to the ditference Ijctween the acceleration of gravity and tlie 

 retardation of friction. In a word, both of these being uniform and inde- 

 pendent of the velocity, then' difference, that is, the eticctive accelej'ating 

 force, down the plane will be uniform and independent of the velocity. 



Such was the reasoiring on which was based all former investigations of 

 the resistance of railway trains, by observing their motions dowir inclined 

 planes. The acceleration due to gravity was calculated ; the actual accele- 

 ration moving down the plane was observed, and the rUtferencc was supposed 

 to give the retarding force due to the resistance. It is evident that by such 

 a mode of proceeding, tlie effect of tlic atmosphere, or of any other cause 

 vvlrich produced a retardation increasing with the speed, was either neglected, 

 or was considered to be of such trifling amount, eoiupared with tlie resistance 

 due to fiiction, that it might be regarded as involved in the estimate of 

 resistance thus obtained with suflicient accuracy for jn-actical purposes. Such, 

 indeed, was the impression on Dr. Lardner's own mind when he coniraenced 

 this investigation, and he accordingly proceeded on the same principles as 

 those adopted by other inquirers, except that in the formida; he included the 

 effect of tlic gyration of the wheels, which was neglected in the calculations 

 of JI. de Pandiour, and which omission entailed an error upon hisresults. 



With a view to determine the actual acceleration of a train down an 

 inclined plane, the Winston and Sutton inclined plains on the Liverjiool and 

 Manchester Railway, and a series of inclines on the (hand Junction Railway, 

 extending from Madeley for several miles towards Crew e, were selected. This 

 figure represents the incUned planes on the Liverpool and Jlauchester Rail- 

 Level. 



IJ miles. 2 miles. U miles, 



way. The siuiuuit level which lies between them is about two miles in 

 length ; the Whiston inchned plane ilescends towards Liverpool, falling at 

 the rate of 1 in 96 for about a mdc and a half, and is succeeded by an in- 

 clination which rises at the rate of 1 in 936 for a considerable distance. 



The Sutton i\iclined plane falls, towards Manchester at the rate of 1 in 89 

 for about a mile and a half, imd is succeeded by a portion of the line nearly 

 level, for a considerable distance towards Manchester. 



The lirst plane on the Urand Junction line descended from the Madeley 

 summit towards Crewe, falling at tlie rate of 1 in 177 for a distance of three 

 miles and a (piartcr ; this is succeeded by another which falls at the rate of 1 

 in 265 for a distance of rather more than tlu'ee miles, which is succeeded by 

 another falhng 1 in 330 for a chstance of nearly a mile and a half. This last 

 is succeeded by a level, wliicli continues for several miles. These planes aie 

 represented in the following thagram. 



Level. 



T7T 563 3^5 



3m. 20clins.no links. 3m. Schns. 721inks. Im. 31c. 311. 

 The 'Wliiston and Sutton inchned planes on the Liverpool and Jlauchester 

 line, are straiglit throughout nearly their whole length. The Madeley iii- 

 fUnes, represented in the diagram, are, in some places, curved with a radius 

 of a mile, tm-ning alternately to the right and to the left; but considerable 

 portions of them are straight. A stake, marked 0, was placed at the summit 

 of each inclined plane, and the length of the )daue descending was divided 

 cut by stakes marked successively, 1, 2, 3, &e., into spaces of one hundred 

 yards. Watches, by wiiich a second could be without difficulty bisected, 

 were provided, and the moment of passing the successive stakes was ob- 

 served to within, at the most, half a second of the truth. Every eare was 

 taken to confer the last degree of accuracy upon these observations ; one 



person w as employed to call out the moment of passing each stake ; another, 

 sui>|ihed with a watch, declared the time, and the third took it down; and 

 in m.iny cases these were checked by having two sets of observers. 



A few cNpeiiments conducted in this manner soon made it manifest that 

 the motion down an inclined plane was not, as has been hitherto supposeil, 

 uniformly accelerated. It was found, for example, that the increase of 

 speed in each successive interval of time was not the same, but was con- 

 tinually less as the motion increased. In other words, the degree of ac- 

 celeration wns t/radualhj dhiimkhed. Now this was an effect evidently indi- 

 cating an increase of the resistance with the increase of speed, and naturally 

 suggested the idea that the atmosphere must have had a more considerable 

 effect than had been supposed. The mathematical formula', comiuoidy used 

 for the determination of resistance, are founded, as has been already stated, 

 on the supposition that the resistance is independent of the speed. These 

 formula" were now applied to the motion of the train down the inclined 

 phuies for short distances, measured from the points at which the trains 

 were respectively started, so that within the range of their apphcation the 

 train might acquire but very little speed, and therefore that the result might 

 lie only slightly affected by the atmosphere. The results of such calculations, 

 applied to the motion of the train for 100, 200, or 300 yards, were found 

 to give a resistance, amounting to from the 400th to the l.jOth of the load. 

 This was not half the amount of the common estimate of the resistance to 

 railway trains, which was about the 250th part of the load, that resistance 

 having been assumed to be the same at all speeds. It occurred to Dr. Lard- 

 ner now to attempt an ajiproximation at the resistance by another process, 

 as follows : — tiains were brought to a level and straight line of raihvay, and, 

 being driven by an engine until they attained a s]ieed of 30 to 35 miles an 

 hour, they were (hsmissed, and allowed to run untU, being gradually re- 

 tarded they were brought to rest. The line being staked out as before the 

 moments of passing, the successive stakes were observed, and the rate at 

 which the train was retarded by the resisting forces was observed, for each 

 100 yards over which it moved; a calculation was made of the amount of 

 resistance by the usual formuhe, founded on the supposition that the resist- 

 ance is independent of the speed; liut these calculations being confined to 

 the first 100, 200, or 300 yards, might be considered as giving a fair ap- 

 proximation, since the change of velocity throughout that distance was not 

 very considerable. The result of such observations indicated a resistance 

 amounting to from a ninetieth to a hundredth of the load. It will be 

 observed that in these last cases the velocity of the train, at wiiich the re- 

 sistance was computed, was very considerable, while, in the former cases, 

 taling the initial motion down an inclined plane, it was very small. The 

 inference, of course, which followed, supposing such calcidations to give 

 correct results, was, that the actual resistance at high speeds was many 

 times more than when the motion is slow. Since, however, these methods 

 of calculation could be regarded as only approximative, and were, in fact, 

 based on principles which were only true on the supposition that the re- 

 sistance was independent of the velocity, which supposition was contracUcted 

 by the residts of the calculations themselves, it was considered necessary to 

 resort to some other and more correct method of determining the resistance. 



If it be admitted that the atmosphere produces any considerable resist- 

 ance, since that resistance must increase in a very high ratio with the speed, 

 it would follow, that if an inclined jilane of sufficient length could be ob- 

 tained, the motion of a train woidd continue to be accelerated until it would 

 olitain a velocity which would produce a resistance from the air, such as. 

 combined with friction, would be equal to the yrovitation down the plaue. 

 Miien sucli a velocity should be attained, the moving force down the plane, 

 being equal to the resisting force, no further acceleration would take ]>lace. 

 As it was thought, however, that the inclined planes, which were accessible, 

 might not be of sufficient length to i)roduce this effect with such trains as 

 it was possible at that time to obtain for experiments, it occiuTed to Dr. 

 Lardner that (he end would be equally attained by staiting the train from 

 the top of the inclined jdane at a considerable speed ; that thus, the ac- 

 celeration it would receive while descending being added to its initial speed, 

 might be expected to give that velocity, at some point of the descent which 

 would be attended by a resistance equal to the gravitation of the train down 

 the plane ; at which point, therefore, acceleration might be expected to 

 cease, and a uniform motion to be maintained to the bottom of the plane. 



The first experiments tried with this view were completely successful, and 

 the result obtained was in exact accordance with what had been anticipated. 

 On the simimit level of the Liverpool and ilanchester Raihvay, marked in 

 the diagram No. 1, a train of foiu' carriages was placed, and was tkawn by 

 an engine to the top of the Wiiston plane {-^), from wiience it was started 

 at a considerable speed. Its motion was accelerated for a short distance, 

 but soon became perfectly unifonii ; and it descended through the greater 

 p.-irt of the plane at the uniform velocity of 31.2 miles an hour. Tins ex- 

 periment was again repeated with the same coaches, increasing the load. 

 .\s was expected from the gravitation of the increased load, a greater velocity 

 was now obtained ; but still a uniform velocity resulted, and the train de- 

 scended the plane with the most perfectly uniform motion, at 33.72 miles 

 an hour. These experiments were tried repeatedly on the same day with 

 the same results. A moderate wind blew down the plane, so that the in- 

 ference was, that this train, in a calm atmosphere, would have suffered a 

 resistance gi-eater than a ninety-sixth part of its weight, at the velocities 

 above mentioncil. This experiment, with a train of four coaches, was re- 

 peated ou the Suttou iilanc, and on the iaeliues new Madeley, represented 



