GEODESY. Ivii 



The azimuths a,, and ao.i, it will be observed, are the astronomical azimuths, 

 or the azimuths which would be determined astronomically by means of an alti- 

 tude and azimuth instrument. 



b. Characteristic property of geodesic line. 



Let a'i.2 = azimuth of geodesic line at Pi, 



a'2.1 = azimuth of geodesic line at P^^ 

 $1, 6., = reduced latitudes of P^ and Po respectively. 



Then the characteristic property of the geodesic line is 



sin ttio cos ^1 = sin (180° — a^.i) cos O2 = cos 60, 



where ^0 is the reduced latitude of the point where the geodesic through Pi and 

 P2 is at right angles to a meridian plane. 



The difference between the astronomical azimuth a^^^ and the geodesic azimuth 

 u'l.o is expressed by the following formula : 



«i.2 — a'1.2 0-^ seconds) = ^V p" ^" {) cos^ <j[> sin 2ai.2> 



where s = length of geodesic line Pi P^, 



a =. major semi-axis of spheroid, 

 ,?=: eccentricity of spheroid, 

 p" =z 2o6264."8, 



^ =z astronomical latitude of Pi, 

 ai,2 = azimuth (astronomical or geodesic) of Pi P2, 



log T2- p"( - J = 7-4244 — 20, for a in feet. 



Thus, for cfi = o and ojo = 45°, for which cos* ^ sin 2ajo = i, the above for- 

 mula gives 



"1.2 — «''i.2 = o-"o74> for s ^ 100 miles, 

 = 0.296, for s = 200 miles, 



so that for most geodetic work this difference is of little if any importance. 



13. Solution of Spheroidal Triangles. . 



The data for solution of a spheroidal triangle ordinarily presented are the 

 measured angles and the length of one side. This latter may be either a geodesic 

 line or a vertical section curve, since their lengths are in general sensibly equal. 

 Such triangles are most conveniently solved in accordance with the rule afforded 

 by Legendre's theorem, which asserts that the sides of a spheroidal triangle (of 

 any measurable size on the earth) are sensibly equal to the sides of a plane 

 triangle having a base of the same length and angles equal respectively to the 

 spheroidal angles diminished each by one third of the excess of the spheroidal 

 triangle. In other words, the computation of spheroidal triangles is thus made to 

 depend on the computation of plane triangles. 



