THE AERODYNAMICS OF A SPINNING SHELL. 
365 
deduced and its first and second differential coefficients obtained from a difference 
formula. In this way we find that initially 
Ej/Ro = 0-000011, 
Rj/Eo = 0-0031, 
for the guns rifled 1 turn in 30 diameters and 40 diameters respectively. Moreover, 
the value of Rj/R 0 diminishes along the trajectory. The neglect of this term is 
therefore justified, provided s > 1-1, and the total error in the solution will probably 
be of the same numerical order. 
The contribution of f (see 3.6235) to the coefficient of rj in equation (3.613) is 
2 kc' 
c (1 + O')’ 
the principal term in this coefficient being — Q 2 /4 s. The relative value of the error 
in omitting this term is therefore 
which is less than 
8 sc'k 
cQ 2 ( 1 ± o-) 
4sat 4s0'i , . 
-Q IT tan 6 
1 • 
The values of these three factors can be obtained from Table VIII. The maximum 
value of this ratio for the 50 degrees trajectory is tan $i- This is negligible. 
In evaluating M to obtain equation (3.6234), terms such as P/Q 2 , kJi/Q 2 , k'/Q 2 are 
neglected. It is unnecessary to evaluate such terms in detail, since it is known that 
k/Q and h/Q are less than 0-02 in all cases. It would, however, be easy to write 
down equation (3.6234) with such terms included. 
4.33. Errors in the Particular Integral. —The errors of the expression for the 
motion of the centre of gravity of the shell, given in (3.632) and (3.633), may be 
obtained from the expansion of the particular integral in powers of l/Q. The ratio 
i6'\/Q0\ of the two terms in ff>, §3.2, can be worked out from the data of the plane 
trajectory. Its initial and greatest value for the gun rifled 1 turn in 30 diameters is 
(0-0008), so that the second term is entirely negligible in comparison with the first. 
Writing therefore <4>j = 6\, it appears that the terms of order l/Q 2 in (3.632) are real 
and so do not affect the drift. The next term is (with y = 0) 
4s 
(jo? 
■n'\+ ( 4 s */]) +fi\ (Ssk— h— k)+vi 
d 
dt 
(4.9/c) +4sk 2 — h< — / 
where ^ (= 4.sff / 1 ) is the coefficient of l/iQ in the first term in the expansion of ?]. 
