THE AERODYNAMICS OF A SPINNING SHELL. 
331 
The second and third components of (3.113) become 
(3.212) y' -\-x9i = k (m—y cos $) — (g/v) cos 0 1 + (yg/v ) (x sin +y cos Of 
(3.213) z f = k ( n—z cos S) + (zg/v) (a? sin (fi + 2 /cos 6j). 
The equation of the plane trajectory corresponding to (3.211) (S = y = 0, x = l) is 
(3.214) v\ = — It (v 1} 0 )/m*—g sin 6 V 
Therefore, if u — v—v 1} u satisfies the equation 
(3.2141) u' — — §) — R(-y l5 g{{x— l) sin 0 x + y cos 0 X }. 
In §§ 4.22, 4.31, we shall show that it is legitimate to regard the value of u determined 
by this equation as zero. We can therefore replace v by v 1 in (3.206), (3.212) and 
(3.213). 
A further discussion shows that (3.212) and (3.213) can be reduced to 
y' = « {ni-y) + {g/v i) y sin 6 U 
zl — k (71 —z) + {gli\) 2 sin 0 lt 
the accuracy and validity of these equations being the same as those of (3.204) and 
(3.205).f These equations combine to give 
d , A qc sin 0, 
dt v x 
or, using the equation of the plane trajectory, 6 \ — —(g/v 1 ) cos 6 U 
(3.215) ^=xr,/c. 
In the cases contemplated this equation is equivalent to (3.212) and (3.213). Then 
(3.215) , (3.206) and the equations of the plane trajectory represent the required 
approximation to the complete equations of motion of the shell. 
In order to convert (3.206) and (3.215) into linear differential equations, it is necessary 
to assume that 71 and k are independent of S, and regard them as functions of v l% 
This approximation involves errors no greater than the previous approximations. If 
Q is treated as a variable, it must be determined by (3.201), T being regarded as a 
known function of the time. All the coefficients in (3.215) and (3.206) are then known 
functions of the time. 
§ 3.3. Equations of Motion of Type f3. 
In the neighbourhood of the vertex of a trajectory of elevation as great as 
70 degrees, the yaw, as stated in § 1.34, may reach large values. In such cases, the 
f With the exception noted in § 4.22. 
