Vibrations of Rifle Barrels. 



341 



Hence, adding these results, we find for the total upward deflection of 

 85'*85, a downward deflection of 65'*25, or finally, a resultant of 20'*6, 

 as the angle which the instantaneous axis makes in an upward direc- 

 tion with the unstrained axis of the barrel, at the moment of the shot 

 leaving the muzzle. 



The course of the shot differs from instantaneous axis of the barrel 

 by an amount depending on the ratio of the transverse linear velocity 

 of the muzzle (due to the vibration) to the muzzle velocity of the shot. 

 The transverse velocity v' of the muzzle consequent on the nth. term 

 vibration in the mth mode, can be obtained by differentiating 9 mn with 

 respect to t, and multiplying by H m (the distance of the nearest node 

 of the mth mode from the muzzle). We then find the ratio v'/v 



= 2 ! B f A f "" (cos 2, * - cos 2, im » ) (25).* 



» m -*-?/? (I ~ Q nm) \ *-m ^m/ 



Computing from this a table of corrections of angle corresponding to 

 Table V representing the alterations of the values of the angles in 

 Table V depending on the vertical linear speed of the barrel, we have 

 approximately 



Table VI. 





I. 



II. 



III. 



1. 



4'-6 up. 



2'* 5 down. 



5'-0 up. 



2. 



0'-35 



o'-o 



o'-o 



3. 



0'-21 



0'-8 



o'-o 



or on the whole 6'*9 of upward inclination must be added to the 20'*6 

 found from Table V, so that the flight of shot lies 27' nearly above the 

 direction of the unstrained axis. 



The actual jump found by experiment for the Lee-Enfield rifle is, I 

 believe, nearly about this amount, but from the uncertainty of the 

 positions found for the nodes in the neighbourhood of the breech, and 

 the small number of terms computed, as well as the doubtful approxi- 

 mation to the pressure curve, no great accuracy could be expected. 

 The example is useful, however, and is introduced to show that the 

 jump depends on the difference between comparatively large quantities, 

 many of which are sure to be varying rapidly with q nm . 



The variations of q nm may be caused either by the variation of T m 

 or t n . For each individual rifle T m of course is constant, depending as 



* It may be noticed that in (24) and (25) sin 2irq nm must = 0, and 

 COS 2irq n m — = dfcl. 



