RELATING TO THE BREAKING OF RAILWAY BRIDGES. 719 



curve the body would describe if it moved along the bridge with an infinitely small velocity. 

 The trajectories corresponding to the four values of /3 contained in the above table are marked 

 by 1, 1, 1, 1 ; 2, 2, 2 ; 3, S, 3 ; 4, 4, 4, * respectively. The dotted curve near B is meant to repre- 

 sent the parabolic arc which the body really describes after it rises above the horizontal 

 line JB*. C is the centre of the right line AB : the curve AeeeB is symmetrical with respect 

 to an ordinate drawn through C. 



17. The inertia of the bridge being neglected, the reaction of the bridge against the body, as 



Cv 

 already observed, will be represented by — , where C depends on the length and stiffness of 



(x — wy ° 



the bridge. Since this expression becomes negative with y, the preceding solution will not be 

 applicable beyond the value of x for which y first vanishes, unless we suppose the body held down 

 to the bridge by some contrivance. If it be not so held, which in fact is the case, it will quit the 

 bridge when y becomes negative. More properiy speaking, the bridge will follow the body, in 

 consequence of its inertia, for at least a certain distance above the horizontal line AB, and will exert 

 a positive pressure against the body : but this pressure must be neglected for the sake of con- 

 sistency, in consequence of the simplification adopted in Art. I, and therefore the body mav be 

 considered to quit the bridge as soon as it gets above the line AB. The preceding solution shews 

 that when /3>|; the body will inevitably leap before it gets to the end of the bridge. The leap 

 need not be high ; and in fact it is evident that it must be very small when ^ is very large. In 

 consequence of the change of conditions, it is only the first maximum value of T which corresponds 

 to the problem, as has been already observed. 



18. According to the preceding investigation, when (i<\ the body does not leap, the tangent 

 to its path at last becomes vertical, and T becomes infinite. The occurrence of this infinite value 

 indicates the failure, in some respect, of the system of approximation adopted. Now tlie inertia 

 of the bridge has been neglected throughout; and, consequently, in the system of the bridge and 

 the moving body, that amount of labouring force which is requisite to produce the ms viva of the 

 bridge has been neglected. If ^, ij be the coordinates of any point of the bridge on the same scale 

 on which ,r, y represent those of the body, and ^ be less than x, it may be proved on the supposition 

 that the bridge may be regarded at any instant as in equilibrium, that 



y \x 1 — xj X' (1 - x) 



(38) 



When X becomes very nearly equal to \, y varies ultimately as (l —x)i'', and therefore i; contains 



terms involving (l - x)' i'', and \-^] i and consequently (— I, contains terms involving 



(1 — w)'''^'. Hence the expression for the vis viva neglected at last becomes infinite; and there- 

 fore however light the bridge may be, the mode of approximation adopted ceases to be legitimate 

 before the body comes to the end of the bridge. The same result would liave been arrived at if 

 ft had been supposed equal to or greater than ^. 



19. There is one practical result which seems to follow from tlie very imperfect solution of 

 the problem which is obtained when the inertia of the bridge is neglected. Since this inertia is 

 the main cause which prevents the tendency to break from becoming enormously great, it would 

 seem that of two bridges of equal length and equal strength, liut une(jual mass, the lighter would 



• The clotted curve ou^ht to have been drawn wholly outnide the full curve. The lw{( curves touch cuch other nt the point 

 where they are cut by the line ACB^ as is reprcACnted in the lif^ure. 



4 Z2 



