THE AERODYNAMICS OF A SPINNING SHELL. 
341 
3.63. The Particular Integral. —-The question of the particular integral is not 
treated generally by Horn and Schlesinger. The former considers shortly a very 
particular case.* Their methods can, however, be extended to obtain the results we 
require. 
We assume an expansion for the particular integral rj } f of the form of 
(3.6261), (3.6262). This integral can be specified in such a way that initially £ = 0, 
i.e., £ <0) = £ (1) = ... = 0, and >/ 0) = 0. It will then be found to be unique. 
On substituting in equations (3.612) and (3.613), with the right-hand side retained, 
and equating powers of iQ, we find first that £ (0) = >/ u) = 0 for all time, and then 
(3.631) ,<*> = 4s0>, f (1) = P isMt/c. 
Jo 
The first two terms in the expansions of rj and £ take the forms 
(8.632) » = Tsr + (S5?{! (4^)+(-r)4 S $ + c' £****/«}; 
(3.633) £ = j"* 
Equations (3.632) and (3.633) will be taken as the standard particular integral. 
Since, moreover, they contain no periodic terms, and the initial value of f is zero and 
those of rj and rj' very small in practice, it is convenient to take this solution as the 
standard solution of the equations of motion in cases where the initial values of >; and 
rj are not exactly known—e.p., in calculating the drift. 
The expansions for rj and f, of which the first two terms are given above, can be 
shown to be asymptotic, but we cannot take up this question here. The numerical 
accuracy of (3.632) and (3.633) will be considered in § 4.33. 
3.64. The general solution of (3.612) and (3.613) may be put in the form 
(3.641) r) — K^+K^ + K^+i,, 
(3.642) £ = + K 2 ^2 + H 3 ^ 3 + 
where K 1; K 2 , K 3 are arbitrary complex constants and rj lf ... , &, ... , have the values 
determined in the last section. 
The particular integral >?, £ represents the motion in an actual trajectory in which 
f is initially zero, and >? and rj start with what may be called their equilibrium values, 
which are numerically very small. The solution (K 3 ^ 3 + ^), (K 3 £ 3 +f) represents the 
motion, of the same type, in a trajectory whose initial tangent makes an angle 
(determined by K 3 ) with the initial tangent of the chosen plane trajectory. This can 
* Loc. cit., p. 340. We hope to publish these extensions in another place. 
