Motion of a Spinning Projectile, 379 



however, give a sudden displacement of the mean position of 



the axis. This is due to our assumption that the law of 



resistance changes suddenly when u = u x . We need have no 



scruples about the sudden shift of the mean position, for it 



is a possibility even if it does not happen, and whatever 



errors are introduced into our deflexions of the axis occur 



only in the neighbourhood where ^ = 1^. A similar thing 



actually does happen at the beginning of the motion, for t 



is not zero when t is zero, and therefore the deflexions given 



by (116) are not zero ; consequently, as soon as the shot 



leaves the gun the mean position of: the axis is suddenly 



shifted to the left (for left-handed spin) and upwards from 



the direction of motion. If there is a side wind at the same 



time, the mean position has, in addition, a sudden displace- 



id 

 ment by an amount — towards the side from which the wind 



u 



is blowing. 



86. When u is just less than u u which is about \u Q for the 

 Mark VI. bullet, we get, using the results in (119), 



= 0'-128, 

 fl 



-<- (?) 



x 9 ° 23' 



= 2 / -30. 

 Also the range is 



X = 600 yards, 

 and, from the ' Musketry Regulations,' 

 a = 36'-7-f51' = 87'-7. 



For a 3-inch solid shot similar to the Mark VI. bullet, 

 I is about ten times as great as for the bullet, and, since I is 

 a factor in the expressions for (x — e), (y — y), X, and a, when 

 these are expressed as functions of u, it follows that corre- 

 sponding values of these four quantities have only to be 

 multiplied by 10 to make them apply to the larger shot. 

 Thus, when the velocity of the 3-inch solid shot has dropped 

 from 2120 to 1060, the first of these being assumed to be 

 the muzzle velocity, the deflexions of its axis are 



#-e=l'-28, 

 y-r, = 23', 

 and at the same time 



X= 6000 yards, 

 «= 6° 7' + 8° 30':= 11° 37'. 



