THE AERODYNAMICS OF A SPINNING SHELL. 
337 
been investigated in general terms by Horn and Schlesinger.* A method, which 
is slightly different algebraically, is more convenient here; the asymptotic properties 
of our solutions, however, may be regarded as established by the researches of these 
authors. 
The equations (3.611) and (3.612) are a pair of linear differential equations with 
respect to the time for the two dependent variables >/ and £, (3.611) being of the 
second order and (3.612) of the first. There must, therefore, be three independent 
solutions. 
It is convenient to eliminate and from (3.611) by the use of (3.612), the result 
being 
(3.613) rj"—{iQ — h —/c x ) r\ — {Q 2 /is + 'iQ (ki — y)—— K \~~ K ic'/ c } >i 
— {iQc'—he'—c"} £ — iQQ?, 
where k x is written for (k— ?’Nx). It is believed that NX is small compared with k, 
so that for simplicity the term NX will usually be omitted in subsequent work. The 
term y will however be retained. 
3.62. The Complementary Function .— A first approximation to the three inde¬ 
pendent complementary functions is obtained, following Horn and Schlesinger, by 
making the substitution, 
i — e y, f = e ?, 
and treating ij and £ as constants in determining >/ and £'■ We also neglect all but 
the highest order terms in 0 in each equation. The equations then reduce to 
(3.621) {-Q 2 x' 2 + Q 2 x'-Q 2 /4:s)rj-iQc^ = 0; 
s 
(3.622) —Krj/c+iQx'£ = 0. 
** 
On eliminating rj and £, and retaining only the terms of highest order in Q, these 
reduce to 
x' (x' 2 —x'+ l/ls) = 0, 
a cubic equation for x! whose three roots correspond to the three independent 
* J. Horn, ‘ Mathematisclie Annalen,’ vol. 52, p. 271 and p. 310. L. Schlesinger, ibid., vol. 63, 
p. 277; ‘Comp. Rend.,’ vol. 142, p. 1031. The investigations of the complementary function given by 
these writers are fairly complete, the asymptotic nature of the expansions being established. The latter 
writer considers a system of n linear differential equations. A similar treatment of the complementary 
function and the particular integral of a special equation is suggested (without proof) by M. de Sparre 
1 Atti (Rendiconti) della R. Acc. dei Lincei,’ 1898, Ser. V., vol. 72, p. Ill ; this writer was obviously lee 
to the solution he gives by his researches on the motion of spinning projectiles. 
[Note added July 30, 1920. See also G. D. Birkhoff, ‘Trans. Amer. Math. Soe vol. 9, p. 219.] 
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