228 MEMOIK OF EATON HODGKINSON. 



Willis has given the following dynamical equation, from wliicli tlie trajectory of 

 the curve described by the moving load may be computed : 



^_ 9 fjct^ y 



clx'~Y^ V'S {2ax—x^f' 

 y and x are the rectangular co-ordinates of the moving weight, the origin being 

 at the extremity of the beam ; y is vertical and x horizontal. 



V=the velocity of the moving weight. 



2a=the length of the beam. 



.</=tlie force of gravity. 



S=the central statical deflection. 

 This equation, and the reasoning by which it is established, accidentally fell 

 into my hands during the time the commissioners were considering it, and in a 

 letter to the secretary. Captain Galton, I pointed out the hypothetical principle 

 on which the equation is founded. This principle is, that the reaction between 

 the moving weight and the beam is equal to the weight which would be neces- 

 sary to deflect the beam, when placed on it at rest, as much as the travelling 

 load deflects it. This position is purely hypothetical, which may or may not 

 give results approximating to the truth, according to the dimension of the quanti- 

 ties which constitute the fixed data of the problem. It is not improbable that 

 this reaction, the amount and direction of which influence the motion of the 

 moving weight over the beam, is continually vibrating between a maximum 

 amount and zero, and that many times during the passage of the weight over the 

 beam the reaction may be nothing, and therefore the moving load be abandoned 

 to the influence of its own gravity only. However this may be, it is certain that 

 its amount is never accm-ately measured by a formula which produces an accelerat- 

 ing force of 



ga' ^ y 



S (;zax-x^f^ 

 as given by Professor Willis. 



This subject has received considerable attention from Mr. H. Cox, in a paper 

 entitled '■'■ Dynamical Deflection and Strain of Girders," which is printed in the 

 ''Civil Engineer and Architects' Journal" for September, 1848. It appears 

 that Mr. Cox has established, from the principle of xis liva, that the moving 

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

 deflection, the elasticity of the girders being supposed perfect. Professor Stokes, 

 however, has shown that this conclusion of Mr. Cox is not true ; that among the 

 sources of laboring force Avhich can be employed in deflecting the girders Mr. 

 Cox has omitted to consider the ris viva arising from the horizontal motion of 

 the body, and therefore has been led to an inference which is not correct. The 

 recorded experimental facts connected with the dynamical deflection of bridges 

 and bars of ii-on are given in the report of the commissioners as follows : 



EWELL BRIDGE. 



The span is 48 feet ; the statical deflection produced by the engine and tender, 

 39 tons, and weight of half bridge 30 tons, wivs only .215 inch. This deflection 

 Avas increased to .245, with a speed of 37 miles per hour. A speed of 51 miles 

 per hour produced a deflection of .235. 



Greatest dynamical deflection _ 

 Statical deflection "~ ' 



