RELATING TO THE BREAKING OF RAILWAY BRIDGES. 725 



actual trajectory. The problem has already been considered in this manner by others by whom it 

 has been attacked. 



My attention has recently been directed by Professor Willis to an article by Mr. Cox On the 

 Dynamical Deflection and strain of Railway Girders, which is printed in The Civil Engineer 

 and Architects Journal for Septenrber, 184S. In this article the subject is treated in a very 

 original and striking manner. Tiiere is, however, one conclusion at which Mr. Cox has arrived 

 which is so directly opposed to the conclusions to which I have been led, that I feel compelled to 

 notice it. By reasoning founded on the principle of vis viva, Mr. Cox has arrived at the result 

 that the moving body cannot in any case produce a deflection greater than double the central 

 statical deflection, the elasticity of the bridge being supposed perfect. But among the sources of 

 labouring force which can be employed in deflecting the bridge, Mr. Cox has omitted to consider 

 the vis viva arising fiom the horizontal motion of the body. It is possible to conceive beforehand 

 that a portion of this vis viva should be converted into labouring force, which is expended in 

 deflecting the bridge. And this is, in fact, precisely what takes place. During the first part of 

 the motion, the horizontal component of the reaction of the bridge against the body impels the 

 body forwards, and therefore increases the vis viva due to the horizontal motion ; and the labouring 

 force which produces this increase being derived from the bridge, the bridge is less deflected than 

 it would have been had the horizontal velocity of the body been unchanged. But during the 

 latter part of the motion the horizontal component of the reaction acts backwards, and a portion 

 of the vis viva due to the horizontal motion of the body is continually converted into labouring 

 force, which is stored up in the bridge. Now, on account of the asymmetry of the motion, the 

 direction of the reaction is more inclined to the vertical when the body is moving over the 

 second half of the bridge than when it is moving over the first half, and moreover the reaction 

 itself is greater, and therefore, on both accounts, more vis viva depending upon the horizontal 

 motion is destroyed in the latter portion of the body's course than is generated in the former 

 portion ; and therefore, on the whole, the bridge is more deflected than it would have been hai 

 the horizontal velocity of the body remained unchanged. 



It is true that the change of horizontal velocity is small; but nevertheless, in this mode of 

 treating the subject, it must be taken into account. For, in applying to the problem the principle 

 of vis viva, we are concerned with the square of the vertical velocity, and we must not omit any 

 quantities which are comparable with that square. Now the square of the absolute velocity of 

 the body is equal to the sum of the squares of the horizontal and vertical velocities ; and the 

 change in the square of the horizontal velocity depends upon the product of the horizontal velocity 

 and the change of horizontal velocity ; but this product is not small in comparison witli the square 

 of the vertical velocity. 



In Art. 22 I have investigated the changes which we are allowed by the general principle of 

 homogeneous quantities to make in the parts of a system consisting of an elastic bridge and a 

 travelling weight, without aff'ecting the results, or altering anytliing but the scale of the system. 

 These changes are the most general that we are at liberty to make by virtue merely of that general 

 principle, and without examining the particular equations which relate to the particular problem 

 here considered. But when we set down these equations, we shall see that there arc some further 

 changes which we may make without affecting our results, or at least without ceasing to be able to 

 infer the results which would be obtained on one .system from those actually obtained on another. 



in 



In an apparatus recently constructed by Professor Willis, which will be described in detai 

 the report of the commission, to which the reader has already been referred, the travelling weiglit 

 moves over a single central trial bar, and is attached to a horizontal arm which is moveable, with as 

 little friction as possible, about a fulcrum carried by the carriage. In this form of the experiment, 

 the carriage serves merely to direct the weight, and moves on rails quite inde])cndrnt of the trial bar. 

 Vol.. VIII. Paut V. •' A 



