198 MEMOIR OP THE LATE PROF. E. HODGKINSON^ F.R.S. 



mical deflection in iron girders. It is true they have not 

 exhausted the subject, nor divested it wholly of its per- 

 plexity ; but they have gained a positive and useful result, 

 by showing to practical engineers the falsity of their posi- 

 tion when they affirm that dynamical deflection is always 

 less than the statical. I may state in conclusion, that Prof. 

 Willis, by a train of reasoning which depends on the as- 

 sumption of each particle of the beam moving into its 

 position, 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 principle which is purely 

 hypothetical, and the correct determination of which is the 

 chief difficulty in the mathematical discussion of the problem. 

 In the Appendix B to the Commissioners^ Report, Prof. 

 Willis has given the following dynamical equation, from 

 which the trajectory of the curve described by the moving- 

 load may be computed : — 



d'-y g ga"" y 



y and oc are the rectangular coordinates 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. 

 205= the length of the beam. 

 ^=the force of gravity. 

 S=the central statical deflection. 



This equation, and the reasoning by w^hich it is esta- 

 blished, accidentally fell into my hands during the time the 

 Commissioners were considering it ; and in a letter to the 

 Secretary, Capt. 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 



