Then, with z = E, and
2ZE = −{ (G′ − GE) / An}2,
(12)
if II denotes the apsidal angle of ῶ, and T the time of a single beat of the axle, up or down,
| II + | GT | = ∫ z3z2 | √ (−2ZE) | dz | , | |
| 2A | z − E | √ (2Z) |
= ½πf + Kznf K′,
(13)
in accordance with the theory of the complete elliptic integral of the third kind.
Interpreted geometrically on the deformable hyperboloid, flattened in the plane of the focal ellipse, if OQ is the perpendicular from the centre on the tangent HP, AOQ = amfK′, and the eccentric angle of P, measured from the minor axis, is am(1 − f) K′, the eccentricity of the focal ellipse being the comodulus κ′.
A point L is taken in QP such that