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Mr. A. Mallock. 



The late W. E. Metford was, I believe, the first to point out the 

 origin of this deviation, showing by experiment that it was due to the 

 unsymmetrical position which the mass of the stock held as regards 

 the barrel ; and, further, that if the initial direction of the shot passed 

 below the apparent direction of aim when the rifle was held in the 

 ordinary position, the initial direction would be high if the rifle were 

 aimed upside down, and to the right or left if the plane of the stock 

 were horizontal and the stock itself to the left or right of the 

 barrel. 



He showed, in fact, that the initial direction of a shot lay on a cone, 

 whose axis was the axis of the barrel at the instant before the ignition 

 of the powder, and in a plane containing the axis of the barrel and the 

 centre of gravity of the rifle, and he rightly attributed the deviation of 

 the shot to the bending couple acting on the barrel, due to the direc- 

 tion of the force causing the recoil not passing through the centre of 

 gravity of the rifle. 



The object of this paper is to examine this problem of " flip " or 

 " jump," as it is called, from a mathematical point of view, and to show 

 what effect may be expected from given variations either in the length 

 of the barrel, the nature of its attachment to the stock, or the nature 

 of the explosive employed. 



The investigation is not merely a matter of curiosity, but has an 

 important bearing on the accuracy of rifle shooting, and until some 

 method is introduced, not of avoiding " jump," but of suitably regu- 

 lating its variation with the variation of explosive force, I think no 

 great advance will be made on the precision already attained in modern 

 rifles. 



This precision is already considerable, and, roughly speaking, any 

 good modern rifle will shoot with a probable deviation of considerably 

 less than 2' from the intended path. When the results indicated in 

 the course of this paper are considered, it seems wonderful that such 

 accuracy should be possible, and it speaks well for the quality and 

 uniformity of the ammunition that such good shooting should be 

 common. 



The problem of "jump " may be stated mathematically thus : — "An 

 elastic tube, to which a mass is unsymmetrically attached, is subjected 

 for a given time to a couple of arbitrary magnitude. Determine the 

 subsequent motion." To solve this problem we must consider the tube 

 and its attached mass as forming a single system, and examine what 

 are the natural modes of vibration of this system, and what their 

 natural periods. The arbitrary couple must be expressed in an 

 harmonic series as a function of time, and the forced vibration which 

 each term of this series will evoke in the system calculated. 



To represent the initial conditions (namely, that at the moment 

 before the explosion the barrel is at rest and unrestrained), such free 



