THE AERODYNAMICS OF A SPINNING SHELL. 
233 
§ 4. The Solution of the Equations of Motion. 
We shall now solve the equations of motion (1) and (2), assuming that M is of form (3). 
We take first the case of rosette motion, in which zero values of S can occur, so that we 
may assume the initial conditions 
S=0, S'= bQ. 
Eliminating f and writing sin ho = y we get 
M' 2 = O 2 (6 2 (i — 2/ 2 ) —2/ 2 H-2/ 3 (i — 2/ 2 ) (i/^ —4g2/ 2 )}.(5) 
The right-hand side is a cubic in y 2 whose roots are such that it may be written in 
the form 
iy' 2 = 4:qQ 2 (h 2 + y 2 ) [d 2 -y 2 ) {f 2 -y 2 ), ( iff 1 < 1 < l/a 2 ). . . . (G) 
Formulae connecting* h, a and / with b, q and s may be obtained most conveniently 
by putting y 2 = 1, y 2 — 0, and by differentiating with respect to y 2 and putting 
y 2 = 0. The resulting formulae are 
iq(l+h ! )(l-a*)(f 3 -l) = 1,.(7) 
4 qa?h?f‘ = 6%.(8) 
4 q (a 2 / 2 — a 2 h 2 — h 2 f 2 ) = — b 2 + l/s — 1.(9) 
A solution of (6) is obtained by assuming, in the usual notation of Jacobian elliptic 
functions, 
where the constant g and the modulus k of the elliptic functions remain to be determined. 
If we solve (10) for cn 2 u and differentiate, we get 
leading to 
t r a 2 {\—q 2 )yy' 
—sn u cn u dn u u = , 
(' a-gV) 
y 
U 
n 
« 4 (i -g 2 ) 
{a 2 -g 2 y 2 ) ( a 2 -y 2 ) {a 2 (l -k 2 ) + {k 2 -g 2 ) y 2 }. 
Comparing this with (6) we may write u' = ± XQ, X constant, and obtain 
f = cf/g 3 , . 
7.2 _ U 2 (t — k 2 ) 
!> - 7 9 •> 5 ....... 
kr-gr 
q 
X 2 g 2 (kr-g 2 ) 
a^l-g 2 ) 
( 11 ) 
( 12 ) 
(13 
* This a has, of course, no connection with the velocity of sound. 
