37o; 



Mr ROHRS, on THE MOTION OF BEAMS 



Let, as before, m be the mass of an unit's length and breadth of rod, R the reaction 

 upwards at point s = 0. Then remembering the value of R, we have 



'f.^-^'t 



Now if ^ be the extreme deflection at time t, 



3 ^ ' '^ 



3 o { S \ 

 V = - — [as^ 1 according to hypothesis ; 



£_ 



whence ~ ^' = S', 



" -,4 ' 



86'*, 



an 



d ^ = 5 cos 



V^.. 



whence ^ is greatest deflection, and the maximum value of 





— IS Ws — . 



dt a- 



To determine the velocity of an arrow discharged from a how. 



Let ABC be the bow, B the centre, APC the cord, which is supposed perfectly flexible, 

 and always stretched between the points AC and the arrow at P. If the bow be much 



thicker at the middle than at the e"nds, which is usually the case, the amount of displacement 

 of the centre of the cord P will be much more tlian twice that of the ends A, C. In a bow 

 with which I experimented the di^lacement was very nearly four times that of the ends, and 

 it will be assumed* that this ratio is constant during the motion. 



Let then E be the initial place of P before the cord is displaced, jiC perpendicular to BP, 



DE = no, EP = 4a?. 



Let the depression of the centre of gravity' of AB = em, where e is a small fraction. 

 e may be taken about .2 or .3 at the outside; ,2 is, I believe, very near it in the bow I 



• This assumption is only an approximation, for if E 

 coincides witli B, the limiting ratio of DP : DE is 2 : VS. 

 But in a practical formula, regard must be had to quantities 



:-?.._.r\ 



of the second and higher orders ; and the ratio in the text is, I 

 think, sufficiently near the truth to be adapted as the mean of 

 the varying ratio of PE : DE. 



