In the astronomical triangle we may apply certain formulas from Chauvenet’s “ Plane and Spherical Trigonometry ” dealing with differential variations in spherical triangles. Lettering as in Fig. i, and keeping A and c constant, we have (by formula 265).
Equation 1 checks from our knowledge that the position line is perpendicular to the bearing. Equation 3 checks with formula 188, in Muir’s Navigation, where it is deduced at somewhat greater length. Equation 2 is unusual.
A point moving uniformly on a graduated scale is always between zero and half a unit from the nearest division point, averaging J of a unit. Random points average J of a unit from the nearest graduation, and, extending the reasoning, indeterminate quantities like t, 8 and L will have an error averaging J of the unit to which they are taken.
Z and M are angles of practically anything up to 180° so that their cosines average about 0.62. If seconds are omitted, the error in L and 8 will average J of a minute. Using this for dL and dS in Equations 1 and 2, we have in either case dh = .62 X J = .15 of a mile from the separate errors introduced into li from taking L and 8 to minutes only.