MECHANICS AND USEFUL ARTS. 23 



creased. Thus, for example, when the carriage, loaded to 1,120 

 pounds, was placed at rest upon a pair of cast-iron bars, nine feet long-, 

 four inches broad, and one and a half inches deep, it produced a de- 

 flection of six tenths of an inch ; but when the carriage was caused to 

 pass over the bars at the rate of ten miles an hour, the deflection was 

 increased to eight tenths, and went on increasing as the velocity was 

 increased, so that at thirty miles per hour the deflection became one 

 and a half inches ; that is, more than double the statical deflection. 

 Since the velocity so greatly increases the effect of a given load in 

 deflecting the bars, it follows that a much less load will break the bar 

 when it passes over it than when it is placed at rest upon it, and, ac- 

 cordingly, in the example above selected, a weight of 4,150 pounds is 

 required to break the bars, if applied at rest upon their centres ; but a 

 weight of 1,778 pounds is sufficient to produce fracture, if passed over 

 them at the rate of thirty miles an hour. 



" It also appeared that, when motion was given to the load, the 

 points of greatest deflection, and, still more, of the greatest strains, 

 did not remain in the centre of the bars, but were removed nearer to 

 the remote extremity of the bar. The bars, when broken by a travel- 

 ling load, were always fractured at points beyond their centres, and 

 often broken into four or five pieces, thus indicating the great and un- 

 usual strains they had been subjected to. 



" We have endeavoured to discover the laws which connect these 

 results with each other and with practice, and for this purpose a 

 smaller and more delicate apparatus was constructed to examine the 

 phenomena in their simplest form, namely, in the case of a single 

 weight traversing a light elastic bar. For the weight in its passage 

 along the bar deflects it, and thus the path of trajectory of the centre 

 of the weight, instead of being a horizontal straight line, as it would 



O ' O O ' 



be if the bar were perfectly rigid, becomes a curve, the form of which 

 depends upon the relation between the length, elasticity, and inertia 

 of the bar, the magnitude of the weight, and the velocity imparted to 

 it. If the form of this curve could be perfectly determined in all 

 cases, the effects of travelling loads upon bars would be known ; but, 

 unfortunately, the problem in question is so intricate that its complete 

 mathematical solution appears to be beyond the present powers of an- 

 alysis, except in the simplest and most elementary case, namely, in 

 which the load is so arranged as to press upon the bar with one point 

 of contact only, or, in other words, the load is considered as a heavy 

 moving point. In practice, on the contrary, a single four-wheeled 

 carriage touches each rail or girder in two points, and a six-wheeled 

 engine, with its tender, has five or six points in contact on each side. 

 This greatly complicates the problem. 



" The above smaller apparatus is so arranged as to comply with the 

 simple condition that the load shall press upon one point only of the bar, 

 and is also furnished with a contrivance by which the effects of vari- 

 ous proportions of the mass of the bar to that of the load can be exam- 

 ined. From the nature of the problem, it is convenient to consider, 

 in the first place, the forms of the trajectories that are described, and 

 the corresponding deflections of the bar, when the mass of the bar is 



