remarkable that the line B was not included in Fraunhofer's second and more accurate determination of the wave lengths, and that the proposed correction to Bλ is about the same, both as to sign and magnitude, as one would have guessed from Fraunhofer's own corrections of the other wave lengths, obtained from his second series of observations.

[A map of the spectrum laid down according to the values of λ instead of λ refers equally to a natural standard, that is, one independent of the material of any prism, and is much more convenient for comparison with spectra obtained by dispersion, not diffraction.]

S. II.

12

[From the Transactions of the Cambridge Philosophical Society, Vol. VIII.

p. 707.]

DISCUSSION OF A DIFFERENTIAL EQUATION RELATING TO THE BREAKING OF RAILWAY BRIDGES.

[Read May 21, 1849.]

To explain the object of the following paper, it will be best to relate the circumstance which gave rise to it. Some time ago Professor Willis requested my consideration of a certain differential equation in which he was interested, at the same time explaining its object, and the mode of obtaining it. The equation will be found in the first article of this paper, which contains the substance of what he communicated to me. It relates to some experiments which have been performed by a Royal Commission, of which Professor Willis is a member, appointed on the 27th of August, 1847, "for the purpose of inquiring into the conditions to be observed by engineers in the application of iron in structures exposed to violent concussions and vibration." The object of the experiments was to examine the effect of the velocity of a train in increasing or decreasing the tendency of a girder bridge over which the train is passing to break under its weight. In order to increase the observed effect, the bridge was purposely made as slight as possible: it consisted in fact merely of a pair of cast or wrought iron bars, nine feet long, over which a carriage, variously loaded in different sets of experiments, was made to pass with different velocities. The remarkable result was obtained that the deflection of the bridge increased with the velocity of the carriage, at least up to a certain point, and that it amounted in some cases to two or three times the central statical deflection, or that which would be produced by the carriage placed at rest on the middle of the bridge. It seemed highly desirable to investigate the motion mathematically, more especially as the maximum deflection of the bridge, considered as depending on the velocity of the carriage, had not

been reached in the experiments*, in some cases because it corresponded to a velocity greater than any at command, in others because the bridge gave way by the fracture of the bars on increasing the velocity of the carriage. The exact calculation of the motion, or rather a calculation in which none but really insignificant quantities should be omitted, would however be extremely difficult, and would require the solution of a partial differential equation with an ordinary differential equation for one of the equations of condition by which the arbitrary functions would have to be determined. In fact, the forces acting on the body and on any element of the bridge depend upon the positions and motions, or rather changes of motion, both of the body itself and of every other element of the bridge, so that the exact solution of the problem, even when the deflection is supposed to be small, as it is in fact, appears almost hopeless.

In order to render the problem more manageable, Professor Willis neglected the inertia of the bridge, and at the same time regarded the moving body as a heavy particle. Of course the masses of bridges such as are actually used must be considerable; but the mass of the bars in the experiments was small compared with that of the carriage, and it was reasonable to expect a near accordance between the theory so simplified and experiment. This simplification of the problem reduces the calculation to an ordinary differential equation, which is that which has been already mentioned; and it is to the discussion of this equation that the present paper is mainly devoted.

This equation cannot apparently be integrated in finite terms†, except for an infinite number of particular values of a certain constant involved in it; but I have investigated rapidly convergent series whereby numerical results may be obtained. By merely altering the scale of the abscissæ and ordinates, the differential equation is reduced to one containing a single constant B, which is defined by equation (5). The meaning of the letters which appear in this equation will be seen on referring to the beginning of Art. 1. For the present it will be sufficient to observe that B varies inversely as the square of the horizontal velocity of the

* The details of the experiments will be found in the Report of the Commission, to which the reader is referred.

[The integral can be expressed by definite integrals. See Art. 7, and last paragraph but one in the paper.]

body, so that a small value of ß corresponds to a high velocity, and a large value to a small velocity.

It appears from the solution of the differential equation that the trajectory of the body is unsymmetrical with respect to the centre of the bridge, the maximum depression of the body occurring beyond the centre. The character of the motion depends materially on the numerical value of B. When ẞ is not greater than , the tangent to the trajectory becomes more and more inclined to the horizontal beyond the maximum ordinate, till the body gets to the second extremity of the bridge, when the tangent becomes vertical. At the same time the expressions for the central deflection and for the tendency of the bridge to break become infinite. When is greater than 1, the analytical expression for the ordinate of the body at last becomes negative, and afterwards changes an infinite number of times from negative to positive, and from positive to negative. The expression for the reaction becomes negative at the same time with the ordinate, so that in fact the body leaps.

The occurrence of these infinite quantities indicates one of two things either the deflection really becomes very large, after which of course we are no longer at liberty to neglect its square; or else the effect of the inertia of the bridge is really important. Since the deflection does not really become very great, as appears from experiment, we are led to conclude that the effect of the inertia is not insignificant, and in fact I have shewn that the value of the expression for the vis viva neglected at last becomes infinite. Hence, however light be the bridge, the mode of approximation adopted ceases to be legitimate before the body reaches the second extremity of the bridge, although it may be sufficiently accurate for the greater part of the body's course.

In consequence of the neglect of the inertia of the bridge, the differential equation here discussed fails to give the velocity for which T, the tendency to break, is a maximum. When ẞ is a good deal greater than, T is a maximum at a point not very near the second extremity of the bridge, so that we may apply the result obtained to a light bridge without very material error. Let T be this maximum value. Since it is only the inertia of the bridge that keeps the tendency to break from becoming extremely great, it appears that the general effect of that inertia is to preserve the bridge, so that we cannot be far wrong in regarding

1

T, as a superior limit to the actual tendency to break. When ẞ is very large, T1 may be calculated to a sufficient degree of accuracy with very little trouble.

Experiments of the nature of those which have been mentioned may be made with two distinct objects; the one, to analyse experimentally the laws of some particular phenomenon, the other, to apply practically on a large scale results obtained from experiments made on a small scale. With the former object in view, the experiments would naturally be made so as to render as conspicuous as possible, and isolate as far as might be, the effect which it was desired to investigate; with the latter, there are certain relations to be observed between the variations of the different quantities which are in any way concerned in the result. These relations, in the case of the particular problem to which the present paper refers, are considered at the end of the paper.

1. It is required to determine, in a form adapted to numerical computation, the value of y' in terms of x', where y' is a function of x' defined by satisfying the differential equation

the value of y' not being wanted beyond the limits 0 and 2c of x'. It will appear in the course of the solution that the first of the conditions (2) is satisfied by the complete integral of (1), while the second serves of itself to determine the two arbitrary constants which appear in that integral.

The equation (1) relates to the problem which has been explained in the introduction. It was obtained by Professor Willis in the following manner. In order to simplify to the very utmost the mathematical calculation of the motion, regard the carriage as a heavy particle, neglect the inertia of the bridge, and suppose the deflection very small. Let a', y' be the co-ordinates of the moving body, x' being measured horizontally from the beginning of the bridge, and y' vertically downwards. Let M be the mass of the body, Vits velocity on entering the bridge, 2c the length of the bridge, g the force of gravity, S the deflection produced by the