VIII. On the Motion of Beams and thin Elastic Rods. By J. H. Rohrs, M.A. 

 Fellow of the Camhi'idge Philosophical Society. 



[Read April 23, I860.] 



DuiiiNG a residence in Switzerland my attention was occasionally directed to certain speci- 

 mens of its ancient national weapon — the steel cross-bow, which I had seen exposed for 

 sale in the shops of dealers in curiosities, and the idea struck me that some such instrument 

 might be serviceable in experiments on the resistance of the air to projectiles of different 

 forms and specific gravities; besides I was curious to compare the efficiency of the " Arbalete" 

 with that of our English long-bow. Accordingly I had no less than three arbaletes made 

 for me in succession by a very skilful Swiss armurier*, the third arbalete stronger than the 

 second, and the second stronger than the first, but I could not succeed in obtaining anything 

 like the velocity I had anticipated ; in fact, 200 feet a second seemed about the superior 

 limit, a velocity far too low for my purposes. This induced me to investigate ab initio by 

 the aid of analysis the motion of a vibrating bow. In order to simplify the subject, I began 

 by considering the motion of an uniform thin elastic lamina depressed at one end, and then 

 suddenly released; the partial differential equation which 1 obtained in this case is the same 

 as that of Poisson, and of which he has given a solution in the shape of a definite integral, 

 which however does not seem easily available for the particular calculation I was engaged in. 

 The differential equation for the motion of a common long-bow or steel-bow is of course dif-. 

 ferent; it depends on the law of the thickness of the spring, which is much stouter in the 

 centre than at the ends; but as the simpler equation for the motion of an uniform rod admits 

 of such exact and pleasing integrations, and is so suggestive of the general phenomena of 

 motion of vibrating rods, uniform or not, I attacked that in the first instance. A highly 

 interesting problem closely connected with that I have more particularly considered, is the 

 determination of the law of vibration of a Railway Girder under the action of a passing 

 load ; this when the load may be considered as collected at one point, and the mass of the 

 girder is neglected, in comparison with that of the load, leads to an equation derived from 

 simple considerations, but of which the numerical integration or tabulation, when the pres- 

 sure of the load is not considered as approximately constant, is so extremely difficult, that 

 I should not have ventured to attempt it, even if it had not been, as it has, accomplished 

 by Professor Stokes, who, in conjunction with Professor Willis, has well nigh exhausted the 



• M. Tschlimy of Moudon. 



Vol. X. PahtII. 46 



