(2) KM = A d[psi]/dt, KN = CR - A cos [theta] d[psi]/dt
= (C - A)R + A d[phi]/dt
so that in the spherical top, with C = A, KN = A d[phi]/dt.
The velocity of H is in the direction KH perpendicular to the plane
COC', and equal to gMh sin [theta] or An^2 sin [theta], so that if a
point in the axis OC' at a distance An^2 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
(3) An^2 sin [theta] e^{([psi] + 1/2[pi])i}
= iAn^2 sin [theta] e^{[psi]i} = d/dt ([rho]e^{[pi]i})
where [rho]e[varpi]i 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,
(4) d.CH/dt = An^2 sin [theta] sin KCH = An^2 sin [theta] KH/CH,
(5) d.1/2CH^2/dt = A^2n^2sin [theta] d[theta]/dt.
and integrating
(6) 1/2CH^2 = A^2n^2(E - cos [theta]),
(7) 1/2OH^2 = A^2n^2(F - cos [theta]),
(8) 1/2C'H^2 = A^2n^2(D - cos [theta]),
where D, E, F are constants, connected by
(9) F = E + G^2/2A^2n^2 = D + G'^2/2A^2n^2.