THE AERODYNAMICS OF A SPINNING SHELL. 
339 
This equation may be solved asymptotically, by successive approximation, by 
writing* 
R = R 0 (l + R 1 + R a ...), 
where 
R, = 0 (l/M) = 0(l/Q 2 ), R 2 = O (l/12 4 ).... 
We obtain, in succession, the approximations 
R 0 = M-*, 
E, = 
verifying the relation R : = 0(l/Q 2 ). The order of magnitude of Rj in practice will 
be discussed in § 4.32, where it will be shown to be negligible.! We therefore take as 
our two standard solutions 
y x = M -i exp jaj 
y 2 = M _i exp | —i £ M* j» 
giving, for the complementary function of (3.623), 
(3.6234) r, = (Qo-)"*{ 1 + 0(1/12)} {K^ + K^}, 
where K 1; K 2 are arbitrary constants, and P 1( P 2 are given by 
Rn P 2 = i 
[12 (l + cr) + i {h + K± (Jl — k + 2y + N'/N)/<r}] dt. 
This is the form of solution which is used in analysing the jump card experiments, 
and contains all the terms that can be required in practice. 
It is now necessary to examine the effect of the term in £ in (3.613), which has so 
far been omitted. The value of f', obtained from (3.612), corresponding to the first 
solution for t], is 
= (k/c) (Qa)~ h e lFl , 
so that, on integrating by parts to obtain the leading terms, 
£ = 
2/C77! 
icQ (1 + <r) 
* At this point the advantage of oar ad hoc method over more general methods is apparent, as we 
obtain in one step a solution with an error 0 (1/it 2 ), whereas the general method requires two steps. 
t We assume that the numerical value of Ri, the next term in the expansion, is a measure of the error 
in the solution caused by omitting all terms after the first. The expansions for y x and y 2 are known to 
be asymptotic for large values of R, so that the error will be some finite multiple of R lt but the size of the 
numerical factor is unknown. 
