336 



Mr. A. Mallock. 



the axis of the barrel, the forced oscillation which the nth term in the 

 series will evoke in the mth mode of the rifle will be, when expressed 

 as the angle through which some particular part of the system bends 

 during the oscillation, is 



6 = 6 m pdk n 1 sm2?Mr- (19). 



1 - ? nm h 



In this expression m is the angle at the place of observation which the 

 unit couple would cause if acting to produce a displacement of the 

 system in the mth mode (the values of 6 m can be found approximately 

 by statical experiments on bending). 



A n is the numerical coefficient of the nth term of the harmonic 

 series, and 



■/ - or 2k (20). 



hi 



To represent the initial conditions, which are that the moment 

 before the explosion the barrel is at rest and unstrained, it suffices to 

 suppose the co-existence of free oscillations of the system, with phases 

 and amplitudes such as to make the velocity and displacement zero 

 when t = 0. If a and b are the amplitudes of the forced and free 

 vibrations respectively, we have 



a sin 2.7T .- + & sin 2tt~ = (21), 



27T t 27T t 



and — a cos 2tt — + ^ b cos 2tt — = (22), 



<n hi *-m hn 



whence q nm = -~ (23), 



hence the free vibration, which at t = leaves the system at rest, so 

 far as the oscillation excited by the nth term in the mth mode is con- 

 cerned, has q nm times the amplitude of the corresponding forced 

 vibration.* 



It is convenient in the complete expression for displacement to refer 

 to the natural periods of the system, which are constant, rather than 

 to the periods contained in the pressure curve. So, substituting for t n 

 its value T m /q nm , we have for the angular displacement of the system 

 at that time after the explosion (i.e., for the sum of the forced and free 

 vibrations at that time due to the term and mode under consideration) 



* For the purposes of this paper it is not necessary to consider the gradual 

 extinction of the free vibrations, for the number of periods involved is so small, 

 even for the highest component taken into account, that extinction will not mate- 

 rially affect the amplitude. 



