RELATING TO THE BREAKING OF RAILWAY BRIDGES. 733 



figs. 2 and 3, represent the number of quarter periods of oscillation of the bridge which elapse 

 during the passage of the body over it. This consideration will materially assist us in under- 

 standing the nature of the motion. It should be remarked too that q is increased by diminishing 

 either the velocity of the body or the inertia of the bridge. 



In the trajectory 1, fig. 2, the ordinates are small because the body passed over before there was 

 time to produce much deflection in the bridge, at least except towards the end of the body's course, 

 where even a large deflection of the bridge would produce only a small deflection of the body. 

 The corresponding deflection curve, (curve I, fig. 3,) shews that the bridge was depressed, and 

 that its deflection was rapidly increasing, when the body left it. When the body is made to move 

 with velocities successively one half and one third of the former velocity, more time is allowed for 

 deflecting the bridge, and the trajectories marked 2, 3, are described, in which the ordinates are far 

 larger than in that marked 1. The deflections too, as appears from fig. 3, are much larger than 

 before, or at least much larger than any deflection which was produced in the first case while the 

 body remained on the bridge. It appears from Table III, or from fig. 3, that the greatest de- 

 flection occurs in the case of the third curve, nearly, and that it exceeds the central statical 

 deflection by about tliree-fourths of the whole. When the velocity is considerably diminished, the 

 bridge has time to make several oscillations while the body is going over it. These oscillations 

 may be easily observed in fig. 3, and their effect on the form of the trajectory, which may indeed 

 be readily understood from fig. 3, will be seen on referring to fig. 2. 



When q is large, as is the case in practice, it will be sufficient in equation (66) to retain onlv 

 the term which is divided by the first power of q. With this simplification we get 



-O, 25^, 25 . 



y=T6:^ = -8-^^'"9-^-.- (.» 



so that the central deflection is liable to be alternately increased and decreased by the fraction 



25 



— of the central statical deflection. By means of the expressions (Gl), (6y), we get 



Sq 



25 / M' ^S, , , 



— = -55 VtTTS = -"2 ^^—^ (72) 



It is to be remembered that in the latter of these expressions the units of space and time are 

 an inch and a second respectively. Since the difference between the pressure on the bridge and 

 weight of the body is neglected in the investigation in which the inertia of the bridge is considered, 

 it is evident that the result will be sensibly the same whether the bridge in its natural position be 

 straight, or be slightly raised towards the centre, or, as it is technically termed, cambered. The 

 increase of deflection in the case first investigated would be diniinislied by a camber. 



In this paper the problem has been worked out, or worked out a])pr(iximately, only in the two 

 extreme cases in which the mass of the travelling body is infinitely great and infinitely small res|)ect- 

 ively, compared with the mass of the bridge. The causes of the increase of deflection in these two 

 extreme cases are quite distinct. In the former case, the increase of deflection depends entirely on the 

 diff'erence between the pressure on tiie bridge and the weight of the body, and may be regarded as 

 depending on the centrifugal force. In the latter, the effect depends on tlie manner in which the force, 

 regarded as a function of the time, is ap|>lie<l to the bridge. In practical cases the masses of the 

 body and of the bridge are generally comparable with eiuh other, and the two effects are mixed 

 up in the actual result. Nevertheless, if we find that each effect, taken separately, is insensil)le, or 

 so small as to be of no practical importance, we may conclude without much fear of error that 

 Vol.. VIII. Part V. S« 



