THE AERODYNAMICS OF A SPINNING. SHELL. 
229 
over a wider range of values of §. By suitably adjusting X and Y, which define the 
couple, and the initial conditions, a curve showing the variation of S with the time 
can be obtained which agrees closely with observations over a complete half-period, 
so that the above expression for M appears to be adequate up to values of § of 35 degrees.* * * § 
Our original approximation with Y = 0 fails in general when <5'> 10 degrees. The 
observed curves suffice to determine X and Y for each round, and as observations 
were taken for a number of different values of the muzzle velocity, M is determined 
by the experiments over a limited range, as a function of the two variables, v, the 
velocity of the shell, and S. 
In solving the equations of motion it is convenient to express the couple by means 
of the non-dimensional coefficients s and q defined by the equation 
M = BQ2 {l — 4g.s- (1 — cos d)}.(3) 
4zS 
It will appear that the motion with S permanently zero is stable or unstable 
according as s > 1 or s<l. For the rounds here analysed, s lies between 1-06 
and 0-83. 
In expressing the results in a standard form it is convenient to use a different 
non-dimensional coefficient / M , which is independent of the mass, moments of inertia, 
size and velocity of the shell, and depends only on the shape of the shell and the 
non-dimensional variables v/a and S, where a is the velocity of sound. This is defined 
by the equationf 
M = pv 2 r 3 sin d f u (v/a, S), . (4) 
where p is the air-density, r the radius of the shell, and the quantities involved are 
expressed in consistent units. 
According to (3), / M is practically constant so long as 8 < 7 degrees, and the value 
of / M (v/a, 0) is strictly comparable with similar values obtained in (A) by analysis 
of the stable rounds on the assumption that f u is independent of 8. 
§ 3. Final Results of the Experiment, 
Fig. 1 shows curves of / M (v/a, 0) as a function of v/a for the four types of shell 
corrected for the effect of the cardsJ. They are reproduced without alteration from 
figs. 4 and 5 of (A) and represent the results for the stable rounds. § The values 
derived from the present analysis of the unstable rounds are plotted for comparison ; 
* When the yaw exceeds 30 degrees the fit is less satisfactory {e.g., III., 11-13). 
f Loc. cit., p. 302, equation 1.103. 
j § 10 below. 
§ The curve for type II. is not actually given, but the data for drawing it can be found in (A) (fig. 13, 
p. 352). 
2 K 2 
