whence
| tan β | A | tan α. |
| C |
(7)
| Fig. 81. |
Hence β ≷ α, and the circumstances are therefore those of the first or second case in fig. 78, according as A ≷ C. If ψ be the rate at which the plane HOJ revolves about OH, we have
| ψ = | sin α | ω = | C cos α | ω, |
| sin β | A cos β |
(8)
by § 18 (3). Also if χ̇ be the rate at which J describes the polhode, we have ψ̇ sin (β − α) = χ̇ sin β, whence
| χ̇ = | sin (α − β) | ω. |
| sin α |
(9)