4c^4 = ---------- [([xi]² + [eta]²)([Omega)1² + [Omega]2²) -([Omega]1[xi] + [Omega]2[eta])²] (a² + c²)² _ 4c^4 | / (a² + c²)² a² (a² + c²)\ = ---------- | LM - N² + ( L------------ - M --- - N --------- ) [zeta]² (a² + c²)² |_ \ 2c²(a² + c²) c² 2c² / _ (a² + c²)(9a² - c²) | - ------------------- [zeta]^4 | = Z, (23) 16c^4(a² - c²) _| where Z is a quadratic in [zeta]², so that [zeta] is an elliptic function of t, except when c = a, or 3a. Put [Omega]1 = [Omega] cos [phi], [Omega]2 = -[Omega] sin [phi], d[phi] d[Omega]1 d[Omega]2 (a² + c²) [Omega]2 ------ = -----------[Omega]2 - [Omega]1 --------- = [Omega]²[zeta] - ---------([Omega]1 [xi] + [Omega]2 [eta])[zeta], (24) dt dt dt (a² - c²) a² + c² N + ------- d[phi] (a² + c²) 4c² ------ = [zeta] - --------- · -------------------------, (25) dt (a² - c²) (a² + c²)² M + ------------[zeta]² 2c²(a² - c²) a² + c² _ _ N + -------[zeta]² / [zeta]d a² + c² / 4c² [zeta] d[zeta] [phi] = | ------- - ------- | ------------------------ · --------------, (26) _/ [root]Z a² - c² _/ (a² + c²)² [root]Z M + ------------[zeta]² 2c²(a² - c²) which, as Z is a quadratic function of [zeta]², are non-elliptic integrals; so also for [psi], where [xi] = [omega] cos [psi], [eta] = -[omega] sin [psi]. In a state of steady motion d[zeta] [Omega]1 [Omega]2 ------- = 0, -------- = --------, (27) dt [xi] [eta] [phi] = [psi] = nt, suppose, (28) [Omega]1[xi] + [Omega]2 [eta] = [Omega][omega], (29) d[phi] a² + c² [omega] ------ = [zeta]- ------- -------[zeta], (30) dt a² - c² [Omega] d[psi] 2a² [Omega] ------ = - ------- -------[zeta], (31) dt a² + c² [omega] a² + c² [omega] 2a² [Omega] 1 - ------- ------- = - ------- -------, (32) a² - c² [Omega] a² + c² [omega] / [omega] a² + c² \² (a² - c²)(9a² - c²) ( ------- - ½ ------- ) = -------------------, (33) \ [Omega] a² - c² / 4(a² + c²) and a state of steady motion is impossible when 3a > c > a.