d[psi]0
U = -2a²--------- - [psi]0
d[lambda]
_
M / [oo] Md[lambda]
= --- - | ----------------
abc _/ 0 (a² + [lambda])P
M
= --- (1 - A0), (20)
abc
with the notation
_
/ [oo] abc d[lambda]
A or A_[lambda] = | ----------------
_/[lambda] (a² + [lambda])P
_
d / [oo] d[lambda]
= -2abc--- | ---------, (21)
da²_/ [lambda] P
so that in (4)
M UxA xA_[lambda]
[phi] = ---xA = ------, [phi]1 = -----------, (22)
abc 1 - A0 1 - A0
in (1) for an ellipsoid.
The impulse required to set up the motion in liquid of density [rho]
is the resultant of an impulsive pressure [rho][phi] over the surface
S of the ellipsoid, and is therefore
_ _ _ _
/ / / /
| | [rho][phi]l dS = [rho][psi]0 | | xl dS
_/_/ _/_/
= [rho][psi]0 (volume of the ellipsoid) = [psi]0 W´, (23)
where W´ denotes the weight of liquid displaced.
Denoting the effective inertia of the liquid parallel to Ox by
[alpha]W´. the momentum
[alpha]W´U = [psi]0W´ (24)
[psi]0 A0
[alpha] = ------ = ------; (25)
U 1 - A0
in this way the air drag was calculated by Green for an ellipsoidal
pendulum.