RELATING TO THE BREAKING OF RAILWAY BRIDGES. 729 



of the quantity within parentheses in (54), we get for the equation to the bridge when at rest 

 under the action of any uniformly distributed force 



') = 5M|?(l-D+f(l -DM — (57) 



If D be the central deflection, >/ = Z) when f = ^ ; so that D : ix :: 25 : l6. 

 Now suppose the bridge in motion, with the mass M travelling over it, and let x, y be the 

 coordinates of M. As before, the bridge would be in equilibrium under the action of the force 



M Ig - Y^j acting vertically downwards at the point whose abscissa is .r, and the system of forces 



rf% . 

 such as M'd^. —-acting vertically upwards at the several elements of the bridge. According 



to the hypothesis adopted, the former force may be replaced by a uniformly distributed force the 

 value of which will be obtained by multiplying by u, and each force of the latter system may be 

 replaced by a uniformly distributed force obtained by multiplying by u', where u is what u 

 becomes when f is put for x. Hence if F, be the whole uniformly distributed force we have 



^^=''[^-'^)-'''C^^^'^^ (-) 



Now according to our hypothesis the bridge must always have the form which it would assume 

 under the action of a uniformly distributed force ; and therefore, if m be the mean deflection at 

 the time t, (57) will be the equation to the bridge at that instant. Moreover, since the point (.r, y) 

 is a point in the bridge, we must have ri = y when ^ = .v, whence y = /lU. We have also 



dV dt' Jo dt- - dt' Jo ^ 126 dt- 



5 Me 

 We get from (55), F^ = -— ^ M- Making these various substitutions in (.08), and replacing 



d d 



— by V — , we get for the diff^erential equation of motion 



— r— 11 = M gn - MV^ u M V (5g) 



2^ * d!c' 126 dx' ^ ' 



Since fj. is comparable with S, the several terms of this equation are comparable with 



Big, Mg, 3TV'S, M'V'S, 



respectively. If then V^ S be small compared with ^, and likewise 71/ small compared witli xV, 

 we may neglect the third term, while we retain the others. This term, it is to be observed, ex- 

 presses the difference between the pressure on the bridge and the weight of the travelling mass. 



V^ S 1 



Since c = i, we have = — -r, which will be small when fi is large, or even moderately large. 



g l6(i 



Hence the conditions under which we are at liberty to neglect the difference between the pressure 

 on the bridge and the weight of the travelling mass are, first, that /3 be large, secondly, that the 

 mass of the travelling body be small compared with the mass of the bridge. If /3 be large, but 

 M be comparable with ISl', it is true that the third term in (59) will be small comjiared with the 

 loading terms; but then it will be comparable witii tlie fourth, and the np|)ro\imiition ado])te(l in 

 neglecting the third term alone would be faulty, in this way, that of two small terms comparable 

 with each other, one would be retained while the other was neglected. Hence, although the ab- 

 solute error of our results would be but small, it would be comparable with the difference between 

 the results actually obtained and those which would be obtained on the supposition that the 

 travelling mass moved with an infinitely small velocity. 



