AND THIN ELASTIC RODS. 361 



If now — be very great, and h very small, A' ;nj | v [ ™"y ^^ neglected in comparison 



o (to y uS J 



with the other terms of the equation in which it stands, and we have finally, 



d'y ch? d*y 



dt^ S da* 



If the rod had not been of constant thickness but of constant breadth, we should have 

 had an equation of the form 



First, for th? motion of an uniform girder under the action of a passing load. 



Let the load be supposed to occupy a small length 2/3 of the girder; 2/3 will be supposed 

 ultimately to vanish. 



Let Q be the pressure distributed over the space 2/3, supposed uniformly so. 



Let v be the velocity of the moving load, supposed uniform, along the girder. 



Then if a be the length of the girder, Y will be made up of two parts mg and V, 

 where F' will = till s = vt, if the load be supposed entering on the girder at time ^ = 0. 



Q 



Y will = — from s = vt, till s = u^ + 2/3, 



2/3 



and Y' will = from thence to the end s = a. 

 Now expanding I^ in a series of sines of between s = and « = a, we have 



Y =^\—- {sm sm n 7r> — sin — 



\.ap la \ a j ] Ttn aJ 



^ /2Q . -rnvt , 7rn3\ 



= 2 — sm sm 



\ a a a J 



in the limit when /3 = 0. 



. . ,„ - cA' 

 Hence writing 6^ for — , 



d-y 



»i -^ = - 6^ -4 + 2 -^ sm sin — + mg. 



df da* \ a a a J 



Let now y = y' Jt y" where y" is the part due solely to the statical deflection of the girder 

 by its own weight. 



d*v" 

 Then = - V ~ + mg ; 

 da* 



d^ da^ a \ a a J 



46—2 



