the disturbances in the initial motion of a shell 319 



most t/ti of the last but one, and may be omitted* ; the equation is 

 then integrable as it stands. If we take as zero time the moment at 

 which the shell begins to be able to rotate freely about the driving 

 band (the moment of emergence of the shoulder), we may take 

 i = for the lower limit of integration and (practically) to ^ 0' 

 Iq - 0. Then we find 



iB^l' + AN I = - im*d {8 {v + itv)}, (5) 



where S {v + iw) denotes the change in v + iw in the interval (0, 0- 

 As the interval in which the shell disengages is small, the action 

 is practically impulsive, so that the term in t,' in equation 5 may be 

 neglected to the present approximation. We thus obtain as a first 

 rough value for I,' , at the moment t when the band clears the muzzle, 



l'{T) = ~'^'^^{h{v + iw)). .(6) 



This can of course be obtained more directly if we are content to 

 ignore the nature of the approximations made at each stage. 

 Since t is of the order of 0-0005 sec. the value of t, (r) is negUgible 

 by comparison. There are of course random gas pressure and blast 

 effects to be superposed on the disturbance resulting from (6), but 

 we are not concerned with these here. The alteration in I,' (t) pro- 

 duced by including the term in ^ can easily be calculated and is 

 found to be of the order of 5 %. 



Thus we see that on this theory the main part of the disturbance 

 will be the acquisition of an angular velocity of the axis t,' (t) 

 given by (6) whose nature depends essentially on 



{v + iw)^. — {v + iw)Q. 



This change of velocity of the free end of the muzzle in the 

 interval (0, t) may a priori be expected to be fairly constant in 

 direction from round to round, thus satisfying the main require- 

 ment of the observations. Whether it could be really proportional 

 to the twist of rifling it is more difiicult to say; owing to the tor- 

 sional strains, such proportionality is not a priori impossible. We 

 would content ourselves with pointing out here that 



{v + iiv)^ — (y + iw)^ 



is directly observable; the principal object of this discussion is to 

 urge the importance of its proper determination. The foregoing 

 discussion also suggests problems in the theory of elasticity. It 

 would be of great interest if any sort of approximation to the 

 elastic vibrations of a gun under firing stresses could be obtained 

 theoretically. 



* This omitted term represents the disturbing effect of the gas pressure acting 

 through the centre of the base. It is only effective after a disturbance has already 

 been set up. 



21—2 



