THE AERODYNAMICS OF A SPINNING SHELL. 
311 
An interesting feature of the damping is that, at a velocity of about 900 f.s., the 
yaw has a distinct tendency to increase {instead of decreasing ) with the time; this 
happens with all four types of shells. Whether this represents a real phenomenon 
or is caused by the impacts on the cards (§ 4.5) is not yet clear. It is not physically 
impossible that ,/ H may be negative for this velocity. These rounds are ignored here, 
and further details must be postponed for Part IV. 
§ 1.3. A Description in General Terms of the Angular Motion of the Axis of a 
Shell, 
The numerical values of f t , f^ and f M described in § 1.21, with the addition of the 
rough values of f n given in § 1.22 make it possible to determine numerically, by the 
principles of rigid dynamics, the motion of a shell projected in any manner, provided 
that the velocity ratio v/a, and the angle of yaw S, do not pass outside the limits for 
which the determination is valid. It is necessary to obtain and solve the dynamical 
equations of motion in terms of the force components before proceeding to the 
inverse process of deducing the forces from the observed motion of the shell. Before 
doing so, however, it is convenient to describe in general terms the motion of the 
shell in various circumstances; this description is qualitative only, and is inserted 
for the purpose of illustration : the quantitative results are reserved for Part IV. 
1.31. The Spinning Top Analogy. —We have already noticed in the Introduction 
the important analogy between the motion of the axis of a shell and the axis of a 
spinning top. With the reservations there made, the analogy is complete, so long 
as/ M can be regarded as independent of S. The equations of motion of a stable shell, 
given in § 3.2, are a generalisation of the equations for the small oscillations of a top in 
the neighbourhood of the vertical. For the general case of stable or unstable motion 
where the yaw need not be small, some use can be made of the exact equations of 
motion of the top (§ 3.4). 
In particular, the condition for the stability of a shell is identical with the 
condition for a top. The condition that the shell should be in stable equilibrium 
with its axis parallel to its direction of motion is that 
(1.311) A 2 N 2 >4B y, 
where A and B are, respectively, the moments of inertia of the shell about longitudinal 
and transverse axes through the centre of gravity, N is the spin of the shell about its 
(longitudinal) axis in radians per second, and /u. sin § is equal to M, the moment of 
the air forces about the centre of gravity. It is therefore convenient to define a new 
variable s, “ the coefficient of stability,” by the equation 
(1.312) s = A 2 N 2 /4B m . 
When s is greater than unity by a sufficiently large amount, a possible form of 
2 x 
VOL. CCXXI.-A. 
