which integrated gives ydx - xdy = Cds. This again may be put in the form r sin a = C, where a is the azimuth of the geodetic at any point--the angle between its direction and that of the meridian--and r the distance of the point from the axis of revolution. From this it may be shown that the azimuth at A of the geodetic joining AB is not the same as the astronomical azimuth at A of B or that determined by the vertical plane A[alpha]B. Generally speaking, the geodetic lies between the two plane section curves joining A and B which are formed by the two vertical planes, supposing these points not far apart. If, however, A and B are nearly in the same latitude, the geodetic may cross (between A and B) that plane curve which lies nearest the adjacent pole of the spheroid. The condition of crossing is this. Suppose that for a moment we drop the consideration of the earth's non-sphericity, and draw a perpendicular from the pole C on AB, meeting it in S between A and B. Then A being that point which is nearest the pole, the geodetic will cross the plane curve if AS be between 1/4AB and 3/8 AB. If AS lie between this last value and 1/2AB, the geodetic will lie wholly to the north of both plane curves, that is, supposing both points to be in the northern hemisphere. The difference of the azimuths of the vertical section AB and of the geodetic AB, i.e. the astronomical and geodetic azimuths, is very small for all observable distances, being approximately:-- Geod. azimuth = Astr. azimuth -(1/12) [e^2/(1 - e^2)] (s^2/[rho]n) (cos^2[phi] sin 2[alpha] + (s/4a)|sin 2[phi] sin [alpha]), in which: e and a are the numerical eccentricity and semi-major axis respectively of the meridian ellipse, [phi] and [alpha] are the latitude and azimuth at A, s = AB, and [rho] and n are the radii of curvature of the meridian and perpendicular at A. For s = 100 kilometres, only the first term is of moment; its value is 0".028 cos^2 [phi] sin 2[alpha], and it lies well within the errors of observation. If we imagine the geodetic AB, it will generally trisect the angles between the vertical sections at A and B, so that the geodetic at A is near the vertical section AB, and at B near the section BA.[3] The greatest distance of the vertical sections one from another is e^2s^3 cos^2 [phi]0 sin 2[alpha]0/16a^2, in which [phi]0 and [alpha]0 are the mean latitude and azimuth respectively of the middle point of AB. For the value s = 64 kilometres, the maximum distance is 3 mm. An idea of the course of a longer geodetic line may be gathered from the following example. Let the line be that joining Cadiz and St Petersburg, whose approximate positions are-- Cadiz. St Petersburg. Lat. 36 deg. 22' N. 59 deg. 56' N. Long. 6 deg. 18' W. 30 deg. 17' E.