584 
ROTATION TOR STABILITY 01 
In order that this quadratic equation should give real values of 
we must have 
r 2 > 
4i?3 co S 3 a %(— - -) 
c 6d" ' c 33 c ll/ 
> 
('-£) 
Suppose the projectile fired from a rifled gun in which the rifling at 
the muzzle would make one turn in n calibres, and suppose /3 the angle 
the rifling at the muzzle makes with the axis of the bore; then 
0 Trd 7 T 
tan p — — = -; 
nd n 
d being the calibre; also 
tan j3 = 
w 
to being the muzzle velocity, and r the angular velocity ; therefore 
2ttw 
Ifc' 
Hence, if the rifling of the gun required to give the requisite spin 
make one turn in % calibres, the calibre being d , we must have 
“sV. 
ClI Ca 
C33C44 ( c n — c 33 ) 
Where, as in practice, the fraction ^, which is equal to the ratio of 
the weight of the air displaced to the weight of the projectile, is so 
small that the squares and higher powers may be neglected, then in 
the formula 
we may put 
but 
and 
c n 
^ c ll c 66 
4<Z 2 C 3 o<?44 (Cq C33) 
fll __ 1. 
Con 
c iz =M P - (a-y), 
C44 = Mkf, Cqq = Mk 9 * 
where /q and Jr, 3 are the radii of gyration of the projectile about an 
equatoreal axis through the centre of gravity and about the axis of 
figure, and therefore we may put 
4a 2 ^ 1 3 — (a —• y) 
This formula shows, therefore, that n remains the same for similar 
projectiles made of the same material, whatever be the calibre. 
