THE AERODYNAMICS OF A SPINNING SHELL. 
239 
motion of the centre of gravity, which function as damping forces as explained in (A).* 
No means of dealing with these effects theoretically has yet been devised for the case 
of large yaw ; the observed changes are clearly of the general type which one’s experience 
of the stable case would lead one to predict. 
§ 7. The Motion in cp. 
It is not difficult to write down a formal expression for the motion in <p. If we take 
equation (25) and substitute for sin ^ from equation (32) we obtain after reduction 
12 . bQ sin 
12 
+ 
^ 1 + cos S 
M2 sin |-/3 
+ 
(l + cos S) y 2 ' 
1 —g 2 sn 2 u 
1 + cos S 1 + cos S (1 —g 2 ) sin 2 1-/3 sir u + sin 2 -g-a cn 2 u 
where b is defined by (30). Now even when a is as big as 30 degrees there is still only 
a maximum difference of 7 per cent, between 1 -j-cos S and 2 , and this maximum is 
only effective for a short part of each period. Hence, for almost all purposes, we are 
still justified in replacing 1 -j-cos $ by 2 f. In order to integrate this equation we notice 
that X12 dt — - du and that 1 = 0 or u = K corresponds to the minimum. Thus, 
<P — 00 ++ 
b sin |-/3 
(1 —g 2 sn 2 u) du 
2 A J u (1 —g 2 ) sin 2 sn 2 u + sin 2 cn 2 
u 
(34) 
This equation contains an elliptic integral of the third kind which can be evaluated 
in 6- functions. We have not, however, made this evaluation or calculated any actual 
0 -curves from (34) mainly because it does nob appear that any further information as 
to the forces acting on the shell would be obtained thereby. We shall content ourselves 
in this paper with a statement of sufficient theoretical results to show that the observed 
^-curves are qualitatively of the form to be expected from (34). A more detailed 
discussion of these curves, however, would, we think, be of some interest. 
It is convenient to treat the motion by using the variable — <p — 0 „—M 2 L 
When /3 is zero, ( p’ will be constant and equal to ^L 2 to our present approximation ; 
with y and as polar co-ordinates the motion then consists of an oscillation in a straight 
line through the origin, for y/ — 0 . In the general case we may eliminate dt between 
equations (33)' and (28), and on substituting for b from (30) obtain an equation for 
dyld\]s in the form| 
I- 
V 
fi + sin 2 
(35) 
If we assume that 7q 2 and f 2 are large compared to sin 2 -|a, the last two brackets 
* In particular see p. 313. 
f It is easy to estimate the precise effect of this approximation in the simple case of the rosette motion. 
The error caused is always very small. 
f In deducing (35) we replace 1-j-cos 3 by 2. 
