I 



318 Mr Foivler and Mr Lock, The origin of 



distance of the centre of gravity from the base point is taken to 

 be d; (2) a constraining couple acting about Ox. 



The velocity of the centre of gravity is therefore V + dh! , and 



F = m*-l{V + (/A'}*. 

 dt 



The moment of this force about the centre of gravity is (? [F . A]. 

 Tlie moment of momentum about the centre of gravity is| 

 ANh. + jB [A . A']. The equation of angular motion is therefore 

 {omitting the constraining coiqile \vhich Ave assume to act about the 

 axis of the bore, Ox) 



I {ANh. + 5 [A . A']} = m^d r| {V -f dM] . A 



= mH ^ [{V + dA'} . A] - m^d [V . A'] , 



or I (.4i\^A + 5i [A . A']} = m*rf [V . A], (1) 



where 5^ = 5 + ni^d^. The y- and s-components of equation (1); 

 are unaffected by the ignored constraining couple. When written, 

 out at length they are 1 



I {ANm + 5i {nr - In')} = nM {w'l - u'n), (2) 



i {ANn + 5i {Im' - ml')) = m-^d {u'm - v'l) (3) 



Equations (2) and (3) are exact. If, now, we recall that the axis of 

 the shell is only slightly inclined to the axis Ox, we may approximate \ 

 by assuming that on and n are small, 1=1, and V = 0. Writing j 

 ^ = ■m + in we can combine these equations into the single one 



j^ {ANt + iB^l'} = - im^d {v' + iw') + im'^du'l ...(4) '■ 



The approximations made in obtaining (4) are certainly legiti- 

 mate. To interpret (4) we must approximate further in a more 

 speculative manner. 



If barrel vibrations are of primary importance, a rough calcu- 

 lation, for the 3-inch shells used in these experiments, shows that 

 I v' + iw' I must not be less than about Jj^u' ; u is the ordinary 

 linear acceleration of the shell at the muzzle. Also | ^ | must be 

 of the order of^| t,' \ r, where t is the total time of emergence of 

 the shell from the barrel (about 0-0005 sees.) so that | ^' | < 0-0015. 

 It follows that the last term in equation 4 is in absolute value at 



* The mass of the shell is '>n*. Vector and scalar products are denoted by 

 [ . ] and ( , ) respeotivelj^. The notation is that of our previous paper, pp. 326, 

 327. t Loc. cit. p. 327. 



