THE AERODYNAMICS OF A SPINNING SHELL. 
333 
or, writing w for /x/AN, 
(3.302) m' — — n (w + 0/ sin 0) — 19', 
(3.303) n' = m (w + f' sin 9) — If' cos 9. 
The corresponding equations of motion of the centre of gravity are 
(3.304) v’ — — B. (v, 8)/m* — g sin 0, 
(3.305) 9' = Kin—(g/v ) cos 9, 
(3.306) f' cos 9 = ten. 
The six equations (3.301) to (3.306) can be solved by a step-by-step process, if R, k, 
/x and T are numerically known functions of v and <5. They are valid without 
restriction as to the size of 8 , and have proved of value for the discussion of 
trajectories at very high elevations. They are, however, necessarily invalid when 
any question of stability is under discussion. 
§ 3.4. Equations of Motion of Type y. 
For the purpose of discussing the initial motion of a shell which is unstable or just 
stable, equations of types a and /3 are invalid, and it is necessary to make use of 
equations corresponding to the equations of energy and angular momentum for a top. 
The equations we shall thus obtain are of far less general applicability than 
types a and /3. 
With this object we take the scalar product of both sides of equation (3.105) into 
the vector [ A . A'] + 0 , and obtain, after reduction, 
(3.401) £B j f |(A') 2 + 2 (0 . [A . A']) + (0) 2 —(A . 0) 2 | = -/* (X . { A'-[A . 0]}). 
Using the axes described in the last section, we note that, over a limited range at 
the beginning of a trajectory, the first two components of 0 are numerically very 
small compared to the third, 9'. We shall find that the effect of 9' itself is negligible 
in the cases we consider. We shall therefore neglect the other components of 0 at 
once. Taking 8 and <p as spherical polar co-ordinates of the axis OA referred to the 
moving axes, so that 
l = cos S, m — sin 8 cos 0, n = sin 8 sin 0,t 
t The angle 0 is not exactly the angle measured by the jump cards, but the difference is negligible. 
The angle 8 is exactly the measured angle of yaw. 
