Along the path of a particle, defined by the C3 of (3),
| sin2 ½θ′ = | y2 | = | y (y − c) | , |
| x2 + y2 | a2 |
(10)
| ½ sin θ′ | dθ′ | = | 2y − c | dy | , | |
| ds | a2 | ds |
(11)
on the radius of curvature is ¼a2/(y − ½c), which shows that the curve is an Elastica or Lintearia. (J. C. Maxwell, Collected Works, ii. 208.)
If φ1 denotes the velocity function of the liquid filling the cylinder r = b, and moving bodily with it with velocity U1,
φ1 = −U1x,
(12)
and over the separating surface r = b