THE AERODYNAMICS OF A SPINNING SHELL. 
.361 
true trajectories; we must now examine more closely the possible effect on the 
drag of the angular oscillations and their rate of damping, by means of the values of 
h and k obtained above. From equations (4.031) and (4.032) it appears that for 
sufficiently large values of t, $ and <p are given approximately by the equations 
<5 = i J (o- 0 /ar)~ e~ {q '~ q -\ 
<P = fo+iPi-Ps), 
so that the shell settles down to a steady precession with the slower precessional 
angular velocity, the yaw gradually diminishing in proportion to the factor 
(o- 0 /o-)® e _(?1_?s) . This quantity is tabulated in column 9, Table VIIIb., on the assump¬ 
tion that y = 0 and h = 3 k. The damping is actually more rapid than is indicated 
by this approximation. 
The question of the rate of damping of the initial oscillations of a shell is of 
importance on account of its effect on the drag R, for the effect, though it may be 
small, will be cumulative, since it tends always to increase R. If it is assumed that 
the effect on R is given by # 
(4.221) R = R 0 (1+M i ), 
where R 0 is a function of v and k is a constant, it is possible to obtain an approximate 
formula for the total change in velocity produced on the assumption that the time 
taken to damp out the oscillations is relatively small. We have 
(4.222) 
m Jo 
— Av = — [ (R—R 0 )c?£, 
<5 2 dt. 
£R„ 
m 
Using (4.031), and integrating on the assumption that o- is constant, q u q 2 , and p 
proportional to t, and j zero, we obtain 
(4.223) 
— Av = 
&J 2 R 0 
2 m 
e 2qi (cosh 2 q.j— cos 2p 2 ) dt, 
_ fcJ J RoO- 0 2 (h 0 + K 0 ) _ _ 
2m {<tq (/; 0 + /c u ) 2 — (Ii 0 — k 0 + 2y 0 ) 2 } 
At present we have no information as to the value of h except at low velocities 
while J varies from round to round so that no numerical results can be given. It 
seems likely that this is a cause of irregularities in range in practice of first class 
importance. The yaw arising from the particular integral Mall also tend to increase 
the resistance, but the effect is of less importance in a low angle trajectory. 
* By symmetry, there can be no odd powers of S in R. 
3 D 2 
