Cambridge Philosophical Society, 239 



sidered apart from diffraction, would be to crowd the planes in the 

 contrary direction, the result seemed decisive in favour of Fresnel's 

 hypothesis, that the vibrations are perpendicular to the plane of po- 

 larization. On the other hypothesis, diffraction would have con- 

 spired with refraction to produce a large crowding in a direction 

 contrary to that in which the observed crowding took place. The 

 amount of crowding, in both positions of the plate, was nearly what 

 would be given by theory, on adopting Fresnel's hypothesis, and 

 supposing that the diffraction took place before the light reached the 

 grooves, but appeared in both cases little less. The difference, how- 

 ever, was comprised within the limits of uncertainty depending upon 

 the errors of observation and the error in the assumed value of the 

 refractive index of the glass plate. 



December 10. — Impact on Elastic Beams. By Homersham Cox, 

 Esq., B.A., Jesus College. 



Among the experiments instituted by the Royal Commission ap- 

 pointed to inquire respecting the use of iron in railway structure, 

 was a series relating to impact on beams. These experiments were 

 undertaken by Professor Hodgkinson, and were conducted in the 

 following manner. The two ends of the beam were fixed in a hori- 

 zontal position, and the blow was given against one of its vertical 

 sides in a horizontal direction. The instrument for giving the blow 

 was a heavy iron ball, hanging down, when at rest, from a point of 

 suspension vertically above the centre of the beam. The ball was 

 raised through different arcs, and after descending by its own gra- 

 vity, struck the beam. The deflection corresponding to different arcs 

 of descent were carefully noted by a graduated scale. 



The object of the present paper is to show that the results might 

 have been predicted by known theoretical principles with consider- 

 able precision and confidence. The problem is divided into two 

 parts: — 1st, to estimate the amount of velocity lost by the ball at 

 the first instant of collision ; 2nd, to ascertain the effect of the elastic 

 forces of the beam in destroying the vis viva which the whole system 

 has immediately after collision. In the first part of the investigation, 

 a general formula, derived from the combination of D'Alembert's 

 principle and that of virtual velocities, is given for the motion of 

 any material system subject to impact. The requisite geometrical 

 condition required for the application of this general formula to the 

 present case is obtained by the assumption, that immediately after 

 impact the form of the beam is a gradual and tolerably uniform curve, 

 such as, for example, the elastic curve of equilibrium. In this way 

 it is determined that about one-half the inertia of the beam is ef- 

 fectively applied at the instant of collision to retard the ball. 



The vis viva of the whole system thus computed is destroyed by 

 the elastic forces of the beam developed by deflection. These, in 

 the second part of the problem, are assutned to vary as the amount 

 of central deflection. By the principle of vis viva a formula is easily 

 obtained, connecting the amount of total deflection with the vis viva 

 of the system immediately after collision. 



Tables are given in which the theoretical and experimental results 



