[phi]1 + [psi]1i = -½iR(x + yi)²a²/(a² + b²), (10)
[phi]2 + [psi]2i = ½iR(x + yi)²b²/(a² + b²), (11)
and for the interspace, the ellipse [alpha] being fixed, and [alpha]1
revolving with angular velocity R
[phi]1 + [psi]1i = -1/8 iRc²sh 2([eta] - [alpha]
+ [xi]i)(ch 2[alpha] + 1)/sh 2([alpha]1 - [alpha]), (12)
[phi]2 + [psi]2i = 1/8 iRc²sh 2([eta] - [alpha]
+ [xi]i)(ch 2[alpha] - 1)/sh 2([alpha]1 - [alpha]), (13)
satisfying the condition that [psi]1 and [psi]2 are zero over [eta] =
[alpha], and over [eta] = [alpha]1
[psi]1 + ½Rx² = 1/8 Rc²(ch 2[alpha]1 + 1), (14)
[psi]2 + ½Ry² = 1/8 Rc²(ch 2[alpha]1 - 1), (15)
constant values.
In a similar way the more general state of motion may be analysed,
given by
w = m ch 2([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (16)
as giving a homogeneous strain velocity to the confocal system; to
which may be added a circulation, represented by an additional term
m[zeta] in w.