MEMOIR' OF EATON HODGKIXSOX. 227 



from D to E. If we examine the effect of these two decomposed forces it will 

 be f'Miiul that the force H cos 0, which is nearly equal to II, since the angle 

 is small, will produce an elongation in the beam A D, and a compression in the 

 beam D B. When the elongation of A D is greater than the compression of 

 D 13, tlie beam between the supports is increased in length ; hence the middle jioint 

 I), where the weight is placed, is moved vertically as well as horizontally into 

 another position. From this force alone the beam would become a wreck if the 

 force II, or the velocity with which the weight is moved from 1) to E, was suffi- 

 ciently large ; but, to prevent this catastrophe, the vertical (-omponent force II 

 sin dimiiiishes the reaction of the weight and beam. The vertical force of the 

 weight, instead of being the weight alone, is now diminished by II sin 0, and is 

 become W — II sin 0, where W is the weight of the movalde body. The effect 

 then of the vertical component is directly opposite to that of the horizontal com- 

 ponent ; and it is evident that under certain conditions either one or the other of 

 these two forces may prevail. Hence the indications of theory are in harmony 

 with the observations of engineers, and fully justify the conflicting evidence 

 which they have given on the subject. Sometimes the conditions of the moving 

 weight and the beam are such as to produce a statical deflection greater than the 

 dynamical, and sometim^.'S the conditions produce a dynamical deflection greater 

 than the statical. The computation of the effects of these component forces is 

 attended ^vith great difliculty, as they bring into full activity the elastic forces 

 of tlie beam and its inertia. The solution, however, of this intricate problem, 

 under certain restrictions, viz., when the weight of the beam is small compared 

 with the moving weight, and the deflection small compared with the length of 

 the beam, has been given by Professor Stokes. (See " Transactions of the 

 Cambridge Philosophical Society," vol. viii, p. 707.) 



The same distinguished analyst has given another solution to this problem 

 when the mass of the moving weight is neglected, or the eflect of the weight 

 reduced to a travelling pressure. The exact solution of the problem lies between 

 these extreme cases, and is therefore circumscribed by the labors of Professor 

 Stokes in such a manner that it can be approached to any degree of proximity 

 required. The general dynamical equations from which the dynamical deflection 

 is computed are so complex that a complete solution of the problem, as exhibited 

 in practice Ijy the moving weight being sustained by the beam in two points, is 

 not likely soon to be furnished. Still, what has been accomplished by Professors 

 "Willis and Stokes is sufficient to show to practical engineers that the startling 

 results of Sir Henry James and Captain Galton, as obtained at Portsmouth, and 

 confirmed on the Ewell and Godstone bridges, arc indicated by dynamical laws, 

 the trutli of whicli cannot be controverted. If this be true — and there can be 

 little doubt of it — no engineer will be justifiedin neglecting a just estimate of its 

 effects on the stability of structures on the safety of which human life depends. 

 The commissioners appointed to inquire into the application of iron to railway 

 stiuctures have rendered essential service to the public by the discovery and 

 experimental development of the difference between statical and dynamical 

 • deflection in iron girders. It is true they have not exhausted the subject, nor 

 divested it wholly of its perplexity ; but they have gained a positive and useful 

 result, by showing to i)ractical engiiieers the falsity of their position when they 

 affirm that dynamical deflection is always less than the statical. _ I may state, 

 in conclusion, that Professor Willis, by a train of reasoning which deixmds on 

 the assumi)tion of each particle of the beam moving into its ixxsition, forming 

 the trajectory, at the same instant of time, has shown that the inertia of the beam 

 is the same as it would be by placing half its weight at the centre. 



This result is derived from a priuci])le which is purely hypothetical, and the 

 correct determination of which is the chief difficidty in the mathematical discus- 

 sion of the problem. In the Appendix 13 to the cJmniissioners' report, Professor 



