330 



Mr. A. Mallock. 



The figure assumed by the muzzle end of the barrel will be nearly 

 exactly the same in each mode as the figure assumed in the corre- 

 sponding mode by an uniform rod whose length is such as to make the 

 distance of the node from its free end equal to the distance from the 

 node to the muzzle of the rifle. 



The couple which acts on the barrel during the explosion is measured 

 by the rate at which the shot is accelerated, the distance of the axis of 

 the barrel from the centre of gravity of the rifle. The effect of a 

 given couple in causing a particular mode of vibration in the barrel 

 depends on its point of application with reference to the nodes of the 

 system as well as on its magnitude. 



Fig. 4. 



— V 



Q.S = l/QQ. PR = j/pQ - Q.T = #Q F . 



CHTKD is the curve inlo which CD is bent by F acting at P. 

 CLTMD is that part of the deformation which belongs to the mode of vibration 

 which has nodes at C and D. 



If in fig. 4, C and D are two adjacent nodes belonging to some 

 particular mode of vibration, it is evident that a couple applied midway 

 between C and D would not cause any displacement of the system in 

 this mode. 



If a is the distance between the nodes C D and a couple pd at point 

 P distant x from c, there will be 



(1) A downward force at C = pd/2x with an equal upward force at P, 

 and 



(2) An upward force at D = - ^ — with an equal downward force 

 at P. 



On the whole, therefore, there is at P an upward force acting 



2 \x a - xj ' 2 \x (a - x )j 



Suppose ijqq = cF to be the displacement which the force F would 

 cause if acting at the point Q, midway between C and D. It is 

 known that if a force F acting at Q causes a displacement y PQ at P, the 

 same force acting at P will cause a displacement y PCl at Q, that is 



* This theorem is due to Lord Rayleigh. 



