THE AERODYNAMICS OF A SPINNING SHELL. 
305 
assumption that L and M are unaffected by the angular velocity of the axis. But 
the values we obtain are too rough to enable us to study the variations of fn with 
any argument. 
1.12. The Effect of the Axial Spin of the Shell. —We have so far ignored the possible effect of the spin N 
of the shell about its axis of symmetry. We shall assume that the preceding components of the force 
system R, L, M and H are not appreciably affected by this spin. This is in accordance with such evidence 
as exists in the case of zero yaw (§ 1.01). If, moreover, the component M were seriously affected by the 
spin, the effect would have been detected by the present trial. No such effect was found (see § 4.13), and 
this fact provides some evidence of the validity of the above assumption, at least as a first approximation. 
The spin N will, however, give rise to certain additional components of the complete force system. 
There will be a couple I which tends to destroy N, and, when the shell is yawed, a sideways force, which 
need not act through the centre of gravity, analogous to that producing swerve on a golf or tennis ball. 
This force must, by symmetry, vanish with the yaw. The swerving force must act normal to the plane 
of yaw, otherwise it would merely have a component which altered R or L (acting in the plane of yaw), 
and we have assumed that no such component exists. The complete effects of the spin N can therefore 
be represented by the addition to the force system of the couples I and J and the force K, acting as 
shown in fig. 3. To procure the correct dimensions we may assume that those components have the 
forms* 
(1.121) I = prNV/i, 
(1.122) J = pvTS-r* sin 8/„ 
(1.123) K = prNr 3 sin 8f K . 
The coefficients / x , / J} / K may depend effectively on a number of variables which we can make no 
attempt to specify in the present state of our knowledge. These components may be expected to be 
very small in comparison with L and M; no certain evidence that they exist is given by our experiments. 
1.13. Relations Between the Components of the Force System. —-The various 
coefficients in the foregoing specifications will all depend on the external shape of the 
shell; results obtained for one shape cannot be applied to another. For shells of 
given shape, however, moving in a given manner, the forces R and L are independent 
of the position of O, the centre of gravity, while the moment M varies with the 
position of 0. If Mi and M 2 are the values of M corresponding to positions O x and 0 2 
of 0, then 
(1.131) Mj = M 2 + 0 1 0 2 (L cos (5+R sin $), 
where C^Oa is positive when Ch is nearer the base than 0 2 . Using the relations 
(l.lOl) to (1.103), and assuming that the yaw is small, the equation (1.131) reduces to 
(1.132) A = fn 2 H--—- (/l +A)- 
r 
This equation is of considerable practical importance, as it enables us to deduce the 
* We shall frequently write F = I/AN, where A is the moment of inertia about the axis of symmetry 
of the shell (see § 1.31). 
2 U 2 
