THE AERODYNAMICS OF A SPINNING SHELL. 
237 
The solution is 
or 
The values of s and q in terms of A, sin \rt, k and g will differ from the values they 
had before by terms of the order sin 2 or h sin kft which are negligible in practice 
compared to af. Hence the previous solution may be applied provided that (32) 
replaces (10) in calculations of the curve of sin as a function of Qt. For convenience 
in computing, (32) may be put in the form (neglecting g 2 ) 
Vi = 
a{ cm u 
1 —g 2 snHT 
(u = K — \Ot), 
•2i* -2i n (sin 2 Ta—sin 2 T/ 3 ) cir n 
sm i^-snrp = 1 - 2 - 
L —g sir u 
(32) 
s ; n i ;; _ sill cos x _ sin sin y 
where 
sm 6 cos 0 
tan 6 = sin |-a cot x/sin |-/3, cos x = cn u. 
Formula (17) remains a valid approximation for 6 2 provided b 2 is defined by (30). 
(33) 
§ 6. A Discussion of the Probable Effects of Damping, and Other Factors Omitted in the 
Foregoing Solution. 
Up to this point we have assumed that the motion in yaw is exactly periodic with 
half-period LiT. This would be exactly true if the couple M were a function of <5 only, 
OP a fixed straight line, and no other couples existed. In actual fact, M is a function 
of the forward velocity and therefore of the time ; OP changes direction under the 
influence of gravity and the cross-wind force, and other couples besides M act on the 
shell, depending on the angular velocity of the shell. All these factors cause progressive 
changes in the curve of yaw from period to period ; for the case of the stable rounds 
they have been discussed at length in (A), where it is shown that they do not appreciably 
affect the determination of M at any velocity for small values of S. In particular, 
the effect of gravity is almost entirely allowed for by using (as we do) the true yaw 
and not the angle between the axis of the shell and some fixed straight line. As explained 
in (A)* the shape of the hole in the cards determines the true yaw and not the angle 
between the axis of the shell and the normal to the card. 
There is no reason to expect that any of these damping effects will be relatively 
more important for an unstable than for a stable shell, except for the change of M with 
the velocity. Although the change in velocity over a single period is always small, 
yet when s is less than or nearly equal to unity a small change in M will cause a fairly 
* Loc. cit., p. 318, footnote. 
2 L 2 
