d[phi] / d[psi] \
----- = ( [psi] + 2(a² + [lambda])-------- )l, (10)
ds \ d[lambda] /
so that over the surface of an ellipsoid where [lambda] and [psi] are
constant, the normal velocity is the same as that of the ellipsoid
itself, moving as a solid with velocity parallel to Ox
d[psi]
U = -[psi] - 2(a² + [lambda])---------, (11)
d[lambda]
and so the boundary condition is satisfied; moreover, any ellipsoidal
surface [lambda] may be supposed moving as if rigid with the velocity
in (11), without disturbing the liquid motion for the moment.
The continuity is secured if the liquid between two ellipsoids
[lambda] and [lambda]1, moving with the velocity U and U1 of equation
(11), is squeezed out or sucked in across the plane x = 0 at a rate
equal to the integral flow of the velocity [psi] across the annular
area [alpha]1 - [alpha] of the two ellipsoids made by x = 0; or if
_
/ [lambda]1 d[alpha]
[alpha]U - [alpha]1U1 = | [psi]-------- d[lambda], (12)
_/ [lambda] d[lambda]
[alpha] = [pi][root](b² + [lambda]·c² + [lambda]). (13)
Expressed as a differential relation, with the value of U from (11),
_ _
d | d[psi] | d[alpha]
--------- | [alpha][psi] + 2(a² + [lambda])[alpha]--------- | - [psi]--------- = 0, (14)
d[lambda] |_ d[lambda] _| d[lambda]
d[psi] d / d[psi] \
3[alpha]--------- + 2(a² + [lambda])-------- ( [alpha]-------- ) = 0, (15)
d[lambda] d[lambda] \ d[lambda]/
and integrating
d[psi]
(a² + [lambda])^3/2 [alpha]-------- = a constant, (16)
d[lambda]
so that we may put
_
/ Md[lambda]
[psi] = | ----------------, (17)
_/ (a² + [lambda])P