236 
MESSRS. R. H. FOWLER AND C. N. H. LOCK ON 
in which we have replaced a by sin \a. These formulae are good approximations in 
practice, when s is not too close to unity. The half-period L>T* (= K/x) tends to 
infinity, but so slowly that no difficulty occurs in practice. Since a, the maximum 
yaw, does not tend to zero with b, the initial disturbance, we have here what may be 
called the unstable case. 
Case (ii). —5> 1 . We now must have /c<45 degrees and, therefore, as cos 2 k>|, 
a ->0 at the same rate as b ; this is the stable case in the usual sense. Equation (16) 
shows further that k -> 0, and, therefore, by (15), X has a definite limit determined 
by 
1-1/s - 4X 2 . 
For given q equations (16)-(20) determine the limiting ratios of b : sin \a. : Jc: g. 
The case in which 5=1 and b ^ 0 can be treated in a similar way. It is found that 
k -> 45 degrees and a and X both tend to zero like ^/b. 
§ 5. Rounds with a Non-Zero Minimum Yaw. 
We shall only consider cases in which the minimum yaw (3 is small, and shall take 
as initial conditions 
$ = (3, S' = 0 , sm S — b\Q. 
The equations of motion become 
<p' sin 2 S— Qbi sin /3 + Q (cos (3 —cos S) = 0 ,.(25) 
S' 2 + cp' 2 sin 2 S —Q 2 &i 2 + I ~d cos (5 = 0 .(26) 
Jj3 B 
If we write yd = sin 2 J<5 — sin 2 1/3, y l vanishes initially and the equation for y x may be 
written 
4 y\~ = IE {bi (cos (3—yd) -y{- b x sin (3 
+ {sm 2 %/3 + y 1 2 )(cos 2 ^(3-y 1 2 )[l/s-4q(sm 2 %/3 + yi)']}. . . . (27) 
We identify (27) with the equation 
Vi 2 = IqQ 2 {hf + yd) {a*-y 2 ) {fi-y/ 2 ), .(28) 
in which ad = sin 2 \ol— sin 2 \(3. Equations (10)—(13) retain their form, and (7)-(9) 
become 
(bi sin ^(3 + cos ^/3) 2 = 4 q (hd + cos 2 j^/3) (J\ 2 —cos 2 ^(3) cos 2 |-a, . . . (29) 
b 2 — (bi cosjf/3— sin^/3 ) 2 = 4,q{h 2 — sin 2 ^) (/ 1 3 + sin 2 -g-/3) sin 2 |-a, . . . (30) 
— b{ + (1 /.s-) cos /3— 1 — 4y sin 2 (2 cos 2 T/3—shr T/3) = 4< q{a 2 f 2 —h 2 a 2 —h 2 f[ 2 ). . (31) 
* T is the time interval between a zero or minimum and an adjacent maximum of the yaw. 
