On a Problem in Navigation

When the distance between the points F (latitude ϕ₁ longitude λ₁ ) and T ( ϕ₂ , λ₂ ) on a rhumb line is considerable, the departure between these points is related to the difference in longitude by the formula Departure = ( λ₂ – λ₁ ) cos l ; l , called the “middle latitude”, is given in terms of the mean latitude $$\frac12(\phi_1+\phi_2)$$ by $$l=\frac12(\phi_1+\phi_2)+c$$ in which c is, in general, a comparatively small calculable angle depending on the co-ordinates of F and T. When the Earth is regarded as a sphere, the method of deriving l , or c , is well known. The question has recently been raised as to the general formula for l , or c , when the spheroidal nature of the Earth is taken into account; this note is intended to supply an answer.