108 
ROTATION OR PROJECTILES. 
and, substituting for F cos a its value (c s w) found in equation (10) in 
text, it results that 
r 2 must not be less than 4 w 2 (l — —^. 
V \ V 
It is to be noted that ^ is the angular velocity with which the plane 
passing through the mean line of flight, or central line of the helical 
path, and a line parallel to the longer axis of the projectile rotates. 
Now, in the right-angled triangle AFC, if AB — the circumference 
of the bore of a gun, and CB = the linear distance of translation in 
which the projectile would, from the rifling, make one revolution with 
a given twist of one in n calibres, then p is the angle which the rifling 
makes at the muzzle with the axis of the bore. Hence 
tan f3 = 
AB izd 7r 
BC nd~~ n 
Also, if AB represent the linear velocity of rotation of a point on the 
exterior of the projectile, and CB represent the linear velocity of 
translation, 
n AB Mr rd 
tan P = — = 2 — = — . 
BC w 2w 
Hence 
7r rd , 2i tw 
= v-, and n — —— . 
n 2w rd 
Now, 
r 2 must not be less than 4w 2 ( 1 
C Q \ 
therefore the minimum value of n is given by the equation 
47 t 2 w 2 
AW 
4>ttV 
~AT 
4w 2 a 
3 c i 
CiC 
16 
4« 2 c 3 c 4 (q — c s) 
(for d = 2a). 
Note the disappearance of the factor w, representing the muzzle velocity. 
