THE AERODYNAMICS OF A SPINNING SHELL. 
345 
where 6 t is the inclination to the horizontal of the tangent to the plane trajectory. 
Since 6 1 < 1 |- degrees, we may replace the latter by 
tan tp = (nx—lz)/(rnx — ly). 
Since is defined by the equation 
y] — ( m—y)+i(n—z), 
we obtain, when $ is sufficiently small, 
t] = sin Se^; 
this expression neglects terms of the second order compared to those retained. It is 
an adequate approximation provided § < 7 degrees. 
The general solution for the equations of type a, given in § 3.65, equations (3.654) 
and (3.6234), is* 
(4.01) , = 
if we ignore, as we may, the particular integral and the third solution, 
write 
Then 
(4.011) 
(4.012) 
Pi = Pi + iqi+P‘j + iq 3 > 
P 2 = p x + iq 1 — (p 2 + iq 2 ). 
Pi — : h\ — |-Q£, 
Jo 
qi = i ( (h+ K ) dt, 
Jo 
P2 — \ 
q* = \ 
[ Qadt, 
Jo 
I" (h—K + 2y) dt/<T , 
Jo 
We shall 
and a 2 = l — l/.s. We observe that p lf p 2 , q 1} q 2 are all nearly proportional to the 
time t. 
The general solution (equation (4.01)) contains two complex arbitrary constants or 
four real constants. By a suitable choice of origin for t and <p these may be reduced 
to two. If the time t — 0 corresponds to a minimum of S and the value 0 = 0 , 
equation (4.01) may be written 
(4.02) >7 = J (<t 0 /ct e m ~ q ' {cosy^ sinh {j—q^+i sin p 2 cosh (j — q 3 )}, 
* Treating N and R as constant, i.e., neglecting the spin reducing couple T. 
t Equation (4.01) reduces approximately to the form 17 = Kt, when s = 1 , and to the form 
V = (<r 0 /o-y {K^’+Ko^ 2 }, when s < 1 , and the shell unstable, the principal parts of <f>i, 0 2 being real and 
positive. The solution then fails completely as an approximation to the actual motion except over a small 
part of the first period. As s approaches the value unity from above, the errors from this cause will begin 
to increase, but the magnitude of these errors can only be estimated by comparison with the solution of 
equations of type y, see § 4.3 below. 
3 b 2 
