bridge, and at the same time restore the general unit of length. If a continue to denote the ratio of the abscissa of the body to the length of the bridge, q will be numerical, and therefore, to restore the general unit of length, it will be sufficient to take the general expression (5) for B. Let moreover be the time the body takes to travel over the bridge, so that 2c Vr; then we get

=

If we suppose & expressed in seconds, and S, in inches, we must put g = 32.2 × 12 = 386, nearly, and we get, g=

Conceive the mass M removed; suppose the bridge depressed through a small space, and then left to itself. The equation of motion will be got from (59) by putting M=0, where M is not divided by S, and replacing M/S by M'/S1, and V. d/dx by d/dt. We thus get

and therefore, if P be the period of the motion, or twice the time of oscillation from rest to rest,

Hence the numbers 1, 2, 3, &c., written at the head of Table III. and against the curves of 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 understanding 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 1, 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 deflection occurs in the case of the third curve, nearly, and that it exceeds the central statical deflection by about three-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 only the term which is divided by the first power of q. With this simplification we get

so that the central deflection is liable to be alternately increased and decreased by the fraction 25/8q of the central statical deflection. By means of the expressions (61), (69), we get

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 diminished by a camber.

In this paper the problem has been worked out, or worked out approximately, only in the two extreme cases in which the mass of the travelling body is infinitely great and infinitely small respectively, 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 difference between the pressure on the bridge and the weight of the body, and may be regarded as depending on the centrifugal force. In the latter, the effect depends on the manner in which the force, regarded as a function of the time, is applied to the bridge. In practical cases the masses of the body and of the bridge are generally comparable with each other, and the two effects are mixed up in the actual result. Nevertheless, if we find that each effect, taken separately, is insensible, or so smail as to be of no practical importance, we may conclude without much fear of error that the actual effect is insignificant. Now we have seen that if we take only the most important terms, the increase of deflection is measured by the fractions 1/8 and 25/8q of S. It is only when these fractions are both small that we are at liberty to neglect all but the most important terms, but in practical cases they are actually small. The magnitude of these fractions will enable us to judge of the amount of the actual effect.

To take a numerical example lying within practical limits, let the span of a given bridge be 44 feet, and suppose a weight equal to of the weight of the bridge to cause a deflection of inch. These are nearly the circumstances of the Ewell bridge, mentioned in the report of the commissioners. In this case, S1 = × 2·15; and if the velocity be 44 feet in a second, or 30 miles an hour, we have T = 1, and therefore from the second of the formulæ (72),

1

The travelling load being supposed to produce a deflection of 2 inch, we have ẞ=127, 1/80079. Hence in this case the deflection due to the inertia of the bridge is between 5 and 6 times as great as that obtained by considering the bridge as infinitely light, but in neither case is the deflection important. With a velocity of 60 miles an hour the increase of deflection '0434S would be doubled.

In the case of one of the long tubes of the Britannia bridge B must be extremely large; but on account of the enormous mass of the tube it might be feared that the effect of the inertia of the tube itself would be of importance. To make a supposition every way

disadvantageous, regard the tube as unconnected with the rest of the structure, and suppose the weight of the whole train collected at one point. The clear span of one of the great tubes is 460 feet, and the weight of the tube 1400 tons. When the platform on which the tube had been built was removed, the centre sank 10 inches, which was very nearly what had been calculated, so that the bottom became very nearly straight, since, in anticipation of the deflection which would be produced by the weight of the tube itself, it had been originally built curved upwards. Since a uniformly distributed weight produces the same deflection as ths of the same weight placed at the centre, we have in this case S1 = 10 = 16; and supposing the train to be going at the rate of 30 miles an hour, we have 7460÷44 10.5, nearly. Hence in this case 25/8q= '043, or nearly, so that the increase of deflection due to the inertia of the bridge is unimportant.

=

In conclusion, it will be proper to state that this "Addition” has been written on two or three different occasions, as the reader will probably have perceived. It was not until a few days after the reading of the paper itself that I perceived that the equation (16) was integrable in finite terms, and consequently that the variables were separable in (4). I was led to try whether this might not be the case in consequence of a remarkable numerical coincidence. This circumstance occasioned the complete remodelling of the paper after the first six articles. I had previously obtained for the calculation of z for values of x approaching 1, in which case the series (9) becomes inconvenient, series proceeding according to ascending powers of 1 − x, and involving two arbitrary constants. The determination of these constants, which at first appeared to require the numerical calculation of five series, had been made to depend on that of three only, which were ultimately geometric series with a ratio equal to §.

The fact of the integrability of equation (4) in the form given in Art. 7, to which I had myself been led from the circumstance above mentioned, has since been communicated to me by Mr Cooper, Fellow of St John's College, through Mr Adams, and by Professors Malmsten and A. F. Svanberg of Upsala through Professor Thomson; and I take this opportunity of thanking these mathematicians for the communication.

[From the Cambridge and Dublin Mathematical Journal, Vol. IV. p. 219

(November, 1849)].

NOTES ON HYDRODYNAMICS.

IV-On Waves.

THE theory of waves has formed the subject of two profound memoirs by MM. Poisson and Cauchy, in which some of the highest resources of analysis are employed, and the results deduced from expressions of great complexity. This circumstance might naturally lead to the notion that the subject of waves was unapproachable by one who was either unable or unwilling to grapple with mathematical difficulties of a high order. The complexity, however, of the memoirs alluded to arises from the nature of the problem which the authors have thought fit to attack, which is the determination of the motion of a mass of liquid of great depth when a small portion of the surface has been slightly disturbed in a given arbitrary manner. But after all it is not such problems that possess the greatest interest. It is seldom possible to realize in experiment the conditions assumed in theory respecting the initial disturbance. Waves are usually produced either by some sudden disturbing cause, which acts at a particular part of the fluid in a manner too complicated for calculation, or by the wind exciting the surface in a manner which cannot be strictly investigated. What chiefly strikes our attention is the propagation of waves already produced, no matter how: what we feel most desire to investigate is the mechanism and the laws of such propagation. But even here it is not every possible motion that may have been excited that it is either easy or interesting to investigate; there are two classes of waves which appear to be especially worthy of attention,