(27)
A sin2 θ dψ/dt = G − G′ cos θ,
(28)
| ψ = ∫ | G − G′z | dt | = ½ ∫ | G − G′ | dt | + ½ ∫ | G + G′ | dt | , | |||
| 1 − z2 | A | 1 − z | A | 1 + z | A |
(29)
the sum of two elliptic integrals of the third kind, with pole at z = ±1; and the relation in (25) (26) shows the addition of these two integrals into a single integral, with pole at z = E.
The motion of a sphere, rolling and spinning in the interior of a spherical bowl, or on the top of a sphere, is found to be of the same character as the motion of the axis of a spinning top about a fixed point.
The curve described by H can be identified as a Poinsot herpolhode, that is, the curve traced out by rolling a quadric surface with centre fixed at O on the horizontal plane through C; and Darboux has shown also that a deformable hyperboloid made of the generating lines, with O and H at opposite ends of a diameter and one generator fixed in OC, can be moved so as to describe the curve H; the tangent plane of the hyperboloid at H being normal to the curve of H; and then the other generator through O will coincide in the movement with OC′, the axis of the top; thus the Poinsot herpolhode curve H is also the trace made by rolling a line of curvature on an ellipsoid confocal to the hyperboloid of one sheet, on the plane through C.
Kirchhoff’s Kinetic Analogue asserts also that the curve of H is the projection of a tortuous elastica, and that the spherical curve of C′ is a hodograph of the elastica described with constant velocity.
Writing the equation of the focal ellipse of the Darboux hyperboloid through H, enlarged to double scale so that O is the centre,