The solution of this equation, which is of the fourth degree, is somewhat tedious; but as the root, which will satisfy the optical conditions of the question, must be the sine of an angle, and necessarily lies between _zero_ and _unity_; and as the protraction, if conducted with due care in the manner already described, affords the means of at once assuming a probable value of ξ not very distant from the truth, the labour of the calculation, in this particular case, is not quite so great as might be expected. But notwithstanding all the abridgments of which the particular case admits, a considerable amount of labour is required, and a corresponding risk of error incurred, in merely introducing the numerical values into the equation preparatory to its solution; and any other method requiring less arithmetical operation, is, of course, greatly to be preferred. I therefore willingly adopted the suggestion of a friend, the benefit of whose advice I have on many occasions experienced, and made use of the following ordinary method of approximating to the root of the equation. If the equation sin ξ - _m_ sin (2  - θ) = 0 (see page 274) be regarded as an expression for the error, when the true value of ξ which would satisfy the equation has been introduced into its first member, we may consider any error in the value of ξ as expressed by the equation: sin ξ - _m_ . sin (2 ξ - θ) = ε and differentiating this expression we have: _d_ ε = cos ξ . _d_ ξ - 2 _m_ cos (2 ξ - θ) . _d_ ξ = {cos ξ - 2 _m_ cos (2 ξ - θ)} . _d_ ξ Then dividing by the differential coefficient we obtain _d_ ε _d_ ξ = --------------------------- 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 -ε _d_ ξ = --------------------------- cos ξ - 2 _m_ cos (2 ξ - θ) -{sin ξ - _m_ sin (2 ξ - θ)} = ---------------------------- cos ξ - 2 _m_ cos (2 ξ - θ) -sin ξ + _m_ sin (2 ξ - θ) _d_ ξ = --------------------------- cos ξ - 2 _m_ cos (2 ξ - 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_.