THE AERODYNAMICS OF A SPINNING SHELL. 
343 
where K x and K 2 are determined by (3.651) and (3.652), and 
(3.655) (, = K 1 f 1 + K 2 f a + K 3 f 8 + [ * rjdt/c, 
Jo 
where by (3.6501), 
(3.656) K 3 = -K, aOo-K, (£,),. 
§3.7. The Solution of Equations of Type y. 
The equations of type /3 are only soluble numerically by step-by-step integration, 
and will not be considered here, but the equations of type y (§ 3.4) reduce,-when p. is 
constant and damping effects are neglected, to the equations of a spinning top, and it 
is convenient to summarize here their solution, in terms of elliptic functions, in the 
form which is most suitable for our purposes. We shall only consider the initial 
conditions § = 0, S' = bQ ; this is the rosette form of motion (§ 1.3) and is usually a 
good approximation to the true motion in its earliest stage. In this case we obtain 
from (3.404) and (3.405) 
(3.701) <p' = 0/(1+ cos $), 
(3.702) S' 2 sin 2 S—Q 2 b 2 sin 2 $+ Q 2 (l — cos <5) 2 — (Q 2 /2s) (l — cos S) sin 2 d = 0. 
If we take Qt as independent variable, the motion depends only on two pai’ameters, 
b and 5. The solution of (3.702) is given by 
(3.703) sin ^<5 = sin \a. cn (K — \Qt, k), 
where a, A, and k are given by the formulae 
(3.7041) = cos j^a cosh ^c, 
(3.7042) b = tan \cl tanh £c, 
(3.7043) tan € = sin rra/sinh Im, 
(3.7044) A = (sin ^a.)/'2k\/s , 
(k = sin e), 
and K is the complete elliptic integral of the first kind to modulus k. Thus the yaw 
oscillates between the values 0 and a, and the value of the period T—the interval 
between successive zeros—is given by 
(3.705) iTT = 2K/A. 
The curve of yaw, <b plotted against Qt is initially concave (convex) upwards, 
when s < 1 (> l). This corresponds to the case of instability (stability) for small 
oscillations. 
3 B 
VOL. ccxxi.— A. 
