Then dividing by the differential coefficient we obtain
d ξ = d ε cos ξ - 2 m cos (2 ξ - θ)
But when ξ becomes ξ + d ξ, ε will also become ε + d ε; but
ε + d ε = 0
therefore d ε = -ε
hence by substitution we have
d ξ = -ε cos ξ - 2 m cos (2 ξ - θ)
d ξ = -{sin ξ - m sin (2 ξ - θ)} cos ξ - 2 m cos (2 ξ - θ)
d ξ = -sin ξ + m sin (2 ξ - θ) cos ξ - 2 m cos (2ξ - θ)
By substituting, therefore, in this last equation the known values of m and θ, and the assumed value of ξ, a correction is obtained, which being applied to ξ and the same process repeated, new corrections may be found until the value of d ξ falls within the limits of error, which may be considered safe in the particular case. I need hardly say, that where so great a body of flame is employed as in the lights of the first order, these limits are soon passed, more especially as one soon acquires by a little experience the means of guessing a value of ξ not very far from the truth. It is this method I have employed in calculating the appended [tables] of the zones, in which I have on all occasions, though, perhaps, with needless exactness, pushed my angular determinations to seconds.