Ivi 



GEODESY. 



log. log. 



cons't 7.54291 — 10 cons't 3.4046 — 10 



Pn 7-32130 Pn 7-3213 



s 5.19818 — 10 s 5.1982 — 10 



15 1. 17609 (15)2 2.3522 



COS <ji 9.88425 — 10 sin 2<ji 9.9934 — 10 



X 1. 12273 y S.zGcj-j — 10 



In. In. 



.»:= 13.266 jV = 0.01861. 



These values of x and y, it will be observed, agree with those corresponding to 

 the same arguments in Table 22. 



When many values for the same scale are to be computed, log j should, of 

 course, be combined with the constant logarithms of (4). Moreover, since in (4) 

 .T varies as AA. and y as (AA.)^, when several pairs of co-ordinates are to be com- 

 puted for the same latitude, it will be most advantageous to compute the pair cor- 

 responding to the greatest common divisor of the several values of AA and derive 

 the other pairs by direct multiplication. 



12. Lines on a Spheroid. 



The most important lines on a spheroid used in geodesy are (a) the curve of a 

 vertical section ; {b) the geodesic line ; and (c) the alignment curve. Imagine two 

 points in the surface of a spheroid, and denote them by J\ and P., respectively. 

 The vertical plane at I\ containing F., and the vertical plane at F^ containing 

 Fi give vertical section curves or lines. The curves cut out by these two planes 

 coincide only when F^ and Fo are in a meridian plane. The geodesic line is 

 the shortest line joining F^ and J\, and lying in the surface of the spheroid. 

 The alignment curve on a spheroid is a curve Avhose vertical tangent plane at 

 every point of its length contains the terminal points F^ and F.,. The curve 

 (ci) lies wholly in one plane, while {b) and {c) are curves of double curvature. 

 In the case of a triangle formed by joining three points on a spheroid by lines 

 lying in its surface, the curves of class {a) give two distinct sets of triangle 

 sides, while the curves of classes (b) and (c) give but one set of sides each. 

 For all intervisible points on the surface of the earth, these different lines differ 

 immaterially in length ; the only appreciable differences they present are in their 

 azimuths (see formula under b below). Of the three classes of curves the first 

 two only are of special importance. 



a. Characteristic property of curves of vertical section. 



Let ai 2 = azimuth of vertical section at Fi through F2, 

 02.1 = azimuth of vertical section at F2 through Fi, 

 Oi, 62 = reduced latitudes of Fi and F.2 respectively, 



81, 80 = angles of depression at F^ and /I respectively of the chord joining 

 these points. 



Then the characteristic property of the vertical section curve joining /'i and F.^ is 



sin ai.2 cos 61 cos \ = sin (a^.i — 180°) cos 62 cos 82- 



