The velocity of H is in the direction KH perpendicular to the plane
COC′, and equal to gMh sin θ or An2 sin θ, so that if a point in the
axis OC′ at a distance An2 from O is projected on the horizontal plane
through C in the point P on CK, the curve described by P, turned
forwards through a right angle, will be the hodograph of H; this is
expressed by
| An2 sin θ e(ψ + 1/2π)i = iAn2 sin θ eψi = | d | (ρeῶi) |
| dt |
where ρeῶi is the vector CH; and so the curve described by P and the motion of the axis of the top is derived from the curve described by H by a differentiation.
Resolving the velocity of H in the direction CH,
and integrating
where D, E, F are constants, connected by
Then
and putting cos θ = z,
| ( | dz | ) = 2n2 (F − z) (1 − z2) − (G2 − 2GG′z + G′2) /A2 |
| dt |
| = 2n2 (E − z) (1 − z2) − (G′ − Gz)2 /A2 = 2n2 (D − z) (1 − z2) − (G − G′z)2 /A2 = 2n2 Z suppose. |
Denoting the roots of Z = 0 by z1, z2, z3, we shall have them arranged in the order
an elliptic integral of the first kind, which with
| m = n √ | z1 − z3 | , κ2 = | z2 − z3 | , |
| 2 | z1 − z2 |
can be expressed, when normalized by the factor √(z1 − z3)/2, by the inverse elliptic function in the form
| mt = ∫ zz3 | √ (z1 − z3) dz |
| √ [4 (z1 − z) (z2 − z) (z − z3)] |
| = sn−1 √ | z − z3 | = cn−1 √ | z2 − z | = dn−1 √ | z1 − z | . |
| z2 − z3 | z2 − z3 | z1 − z3 |
Interpreted dynamically, the axis of the top keeps time with the beats of a simple pendulum of length
suspended from a point at a height 12 (z1 + z3)l above O, in such a manner that a point on the pendulum at a distance
from the point of suspension moves so as to be always at the same level as the centre of oscillation of the top.
The polar co-ordinates of H are denoted by ρ, ῶ in the horizontal plane through C; and, resolving the velocity of H perpendicular to CH,
| ῶ = 12 ∫ | G′ − Gz | dt | = ∫ z3 | (G′ − Gz) / 2An | dz | , | ||
| E − z | A | E − z | √ (2Z) |
an elliptic integral, of the third kind, with pole at z = E; and then
| = tan−1 | A sin θ dθ/dt | = tan−1 | √ (2Z) | , |
| G′ − G cos θ | (G′ − Gz) / An |
which determines ψ.
Otherwise, from the geometry of fig. 4,
| ψ = ∫ | G − G′z | dt | = 12 ∫ | G − G′ | dt | + 12 ∫ | G + G′ | dt | , | |||
| 1 − z2 | A | 1 − z | A | 1 + z | A |
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,
with α2 + λ, β2 + λ, λ denoting the squares of the semiaxes of a confocal ellipsoid, and λ changed into μ and ν for a confocal hyperboloid of one sheet and of two sheets.
then in the deformation of the hyperboloid, λ and ν remain constant at H; and utilizing the theorems of solid geometry on confocal quadrics, the magnitudes may be chosen so that
| λ − μ | = | 1 + z | , | μ − ν | = | 1 − z |
| λ − ν | 2 | λ − ν | 2 |
with z = cos θ, θ denoting the angle between the generating lines through H; and with OC = δ, OC′ = δ′, the length k has been chosen so that in the preceding equations
and δ, δ′, k may replace G, G′, 2An; then
| 2Z | = | 1 | ( | dθ | ) | 2 | = | 4KH2 | , |
| 1 − z2 | n2 | dt | k2 |
while from (33-39)
| 2Z | = | 4 (α2 + μ) (β2 + μ) μ | , |
| 1 − z2 | k2 (μ − λ) (μ − ν) |
which verifies that KH is the perpendicular from O on the tangent plane of the hyperboloid at H, and so proves Darboux’s theorem.
Planes through O perpendicular to the generating lines cut off a constant length HQ = δ, HQ′ = δ′, so the line of curvature described by H in the deformation of the hyperboloid, the intersection of the fixed confocal ellipsoid λ and hyperboloid of two sheets ν, rolls on a horizontal plane through C and at the same time on a plane through C′ perpendicular to OC′.
Produce the generating line HQ to meet the principal planes of the confocal system in V, T, P; these will also be fixed points on the generator; and putting
then
is a quadric surface with the squares of the semiaxes given by HV·HQ, HT·HQ, HP·HQ, and with HQ the normal line at H, and so touching the horizontal plane through C; and the direction cosines of the normal being
the line of curvature, called the polhode curve by Poinsot, being the intersection of the quadric surface (44) with the ellipsoid (46).
There is a second surface associated with (44), which rolls on the plane through C′, corresponding to the other generating line HQ′ through H, so that the same line of curvature rolls on two planes at a constant distance from O, δ and δ′; and the motion of the top is made up of the combination. This completes the statement of Jacobi’s theorem (Werke, ii. 480) that the motion of a top can be resolved into two movements of a body under no force.
Conversely, starting with Poinsot’s polhode and herpolhode given in (44) (46), the normal plane is drawn at H, cutting the principal axes of the rolling quadric in X, Y, Z; and then
this determines the deformable hyperboloid of which one generator through H is a normal to the plane through C; and the other generator is inclined at an angle θ, the inclination of the axis of the top, while the normal plane or the parallel plane through O revolves with angular velocity dψ/dt.
The curvature is useful in drawing a curve of H; the diameter of curvature D is given by
