192 MEMOIR OF THE LATE PROF. E. HODGKINSON, F.R.S. 



bars, when broken by a load passing over tbem, were frac- 

 tured at points beyond their centres, often into four or five 

 pieces, indicating the unusual strains to which they had 

 been subject. From these unexpected results there is no 

 appeal, however much they may be at variance with the 

 impressions of the most gifted engineers. It now remains 

 to connect these results with well-established mechanical 

 laws, a problem of great difficulty, the solution of which 

 has been accomplished by the labours of Prof. Willis and 

 Prof. Stokes (see ' Preliminary Essay on the Effects pro- 

 duced by causing Weights to travel over Elastic Bars,' by 

 the Rev. Robert Willis, F.R.S., &c.). 



By neglecting the inertia of the bar, as being small in 

 relation to the moving weight. Prof. Stokes has shown 

 that 



•?)■ 



D=S + i(- 



D = central dynamical deflection of the bar, produced by 

 the weight moving at the velocity V. 



S = central statical deflection produced by the same 

 weight. 



Z=the length of the bar in feet. 



Hence the dynamical deflection is double of the statical, 

 when the velocity of the moving weight is 'v/2 times the 

 length of the bar between the supports. 



These results were not readily accepted by practical 

 men, as they had been accustomed to connect high velocities 

 of the train with small deflections of the bridge over which 

 it passed. 



The late Robert Stephenson, in his evidence before the 

 Commissioners, states that he had seen the deflections less 

 as the train passed over than when it was in repose. From 

 the observations which he had made, he felt quite satisfied 

 upon the point, that no revision of the practical rules re- 



