THE AERODYNAMICS OF A SPINNING SHELL. 
351 
obtaining values of /3 X ( t ,) and the times of occurrence of the minima with some 
accuracy. Such curves are shown in fig. 12. 
The following quantities have now been determined from the observations, viz. : 
the (assumed constant) value of Q, the times T'j, T' 2 , &c., of the occurrence of the 
minima of S (those are more accurately determined than the times of the maxima), 
and the values of oq ( t ) and (3 X ( t ) over the range of the experiments. These values 
are given in Table V. 
4.11. Derivation of the Various Force Components .—It remains to derive the 
values of the various force components. By equations (4.054), (4.011) 
(4.111) p 2 (T' n ) — p 2 (T' n _j) = 71-, [ Qa-dt = 2t r, 
J TVi 
giving, as a sufficient approximation, 
(4.112) 
- 2ir/T 
(T = T'„—TV.), 
(4.113) 
s = 
1-(2tt/QT) 2 
(4.114) 
, (v\ = A 2 N 2 
M \a) 4B spv 2 r 6 
where <r, s, and v correspond to the time \ (T / n + T / n _ 1 ). T is therefore the time 
between successive minima of S. The values of s and f ^ obtained in this manner, or, 
in a similar way, taking an average over several periods, with the corresponding 
values of /u and vfa, are given in Table VI.,* and provide the data on which figs. 4 and 5 
and Table I. were constructed. 
By comparing the values of f M for shells of form A, with three different positions 
of the centre of gravity, the values of jf L were deduced by the formulae of § 1.13. 
This deduction was done graphically as shown in fig. 13. According to § 1.13 the 
relation between / M and l, the distance of the centre of gravity from the base 
of the shell, should be linear. Fig. 13 shows that all the observed points lie 
on straight lines within the limits of error of the observations. The slope of each 
line determines the value of f L The values of f h are shown plotted against v/a in 
fig. 4. 
* For the rounds fired from the gun rifled 1 in 30 the time of the first minimum near the muzzle is, 
in general, badly determined, and the first period is therefore omitted in determining a mean value for s. 
For the rounds fired from the gun rifled 1 in 40 the time of the first minimum can be determined with 
fair accuracy by extrapolation. 
3 c 
VOL. CCXXI.-A. 
