the disturbances in the initial motion of a shell 317 



progressively with the muzzle velocity and also with the twist of 

 rifling. To this we may add on the evidence of Table I that the 

 size of the disturbance appears to be roughly proportional to O. 



On referring to the characteristics of the disturbances due to 

 the two possible causes described in sections 2 and 3, we see that 

 the evidence is fairly conclusively against the theory of random 

 gas pressure variations which demands random ^q's and Sq' inde- 

 pendent of n. On the other hand there is nothing to prevent its 

 fitting in with the theory of barrel vibrations. We must examine 

 this theory now in greater detail, though, as we have already said, 

 all we can say is speculative and intended only to suggest lines of 

 further work on the problem. 



(5) The motion of a shell with the base constrained. Let us con- 

 sider the circumstances under which a shell emerges from the 

 muzzle of a gun. It seems probable that even for fairly small values 

 of the clearance* the shell will cease to touch the bore, except 

 at the driving band, from the moment when its shoulder reaches 

 the lip of the muzzle. This is a direct consequence of the actual 

 numerical quantities concerned — it is unnecessary to give numerical 

 details here. Thus from the moment at which the shoulder emerges 

 the shell may be regarded as practically free to swing about the 

 driving band|. 



Let us suppose in the first instance that owing to the firing 

 stresses the muzzle end of the gun vibrates, so that the shell is 

 constrained to take up the sideways velocity of the end of the gun 

 just before it leaves the muzzle. We will assume first that the axis 

 of the bore is unaltered in direction by this vibration. To work out 

 the motion of the shell under such conditions we refer it to fixed 

 rectangular axes Ox, Oy, Oz, such that Ox is parallel to the axis of 

 the bore, Oy vertical, and Oz horizontal and to the right as viewed 

 from behind the gun. That point in the shell whose motion is con- 

 strained (the base point) may be taken to be the centre of a section 

 through the middle of the driving band. Let A be a unit vector 

 representing the direction of the axis of the shell (components 

 I, m, n) and V a vector representing the total constrained velocity 

 of the base point (components u, v, iv). Then at the moment when 

 the driving band disengages, u is the muzzle velocity and v, w are 

 the components of the barrel vibration. 



The resultant action on the shell may be taken to be (1) a force 

 F acting through the base point, whose position is defined by the 

 vector — dh. relative to the centre of gravity of the shell. The axial 



* The excess of the ininiimun iiitern:il diameter of the bore over the maximum 

 external diameter of the shell (driving band ignored). 



t If the size of the initial disturbance is found to vary decidedly with the 

 clearance, this paragraph will need modification. But the succeeding arguments 

 hold unaltered provided less time is taken to refer to the moment at which the shell 

 is first free to swing about the band. 



VOL. XX. PART III. 21 



