Iviii GEODESY. 



a. Spherical or spheroidal excess. 

 The excess of a spheroidal triangle of ordinary extent on the earth is given by 



€ (in seconds) = /j" > 



Pm Pn 



where 6" is the area of the spheroidal or corresponding plane triangle ; p„„ p„ are 

 the principal radii of curvature for the mean latitude of the vertices of the tri- 

 angle ; and p" = 206 264."8. For a sphere, p,„ = p„ = radius of the sphere. 



Denote the angles of the spheroidal triangle by A, B, C, respectively ; the cor- 

 responding angles of the plane triangle by a, /3, y (as on p. xviii) ; and the sides 

 common to the two triangles by a, l>, c. Then 



S =^ h ab sin 7 = ^ ^r sin a^=.\ ca sin ^. 



a = A — ^e, /3 = B — le, 7 = C— ^ e. 



Tables 13 and 14 give the values of log (p"/2p„,p„) for intervals of 1° of astro- 

 nomical or geographical latitude.* 



14. Geodetic Differences of Latitude, Longitude, and 



Azimuth. 



a. Primary triangulation. 



Denote two points on the surface of the earth's spheroid by Bi and B> respec- 

 tively. Let 



s = length of geodesic line joining Bi and B.j, 

 <^j, ^2 = astronomical latitudes of Bi and Bj, 

 Ai, /V> = longitudes of /^j and 1^2, 

 AA = Ao — Aj, 



ai.2 = azimuth of jPi Bo (s) at Bi, 

 02.1 = azimuth of Bo B^ (J) at jP,? 

 e = eccentricity of spheroid, 

 p,„, p„ =z principal (meridian and normal) radii of curvature at the point T^. 



Then for the longest sides of measurable triangles on the earth the following 

 formulas will give ^2> ^i ^'^d a^.i iri terms of ^j, Aj, ajo, and s. The azimuths are 

 astronomical, and are reckoned from the south by way of the west through 360°. 



a' = 180° — aj2, and a^.i = 180° -\- a", for ajo <l8o° 



a = ai.2 — 180°, and ao.i = 180° — a", for a.i., > i8o° 



(1; 



" = ,{{.+4 7^= (tocos'* COS'.'} (.) 



'C = i T^r? cos^ </>i sin 2 a' (3) 



* For the solution of very large triangles and for a full treatment of the theorj- thereof, consult 

 Die Mathematischen iind rhysikalischen Thcorieen der Hdhe7-e)i Gcoddsic, von Dr. F. R. Ilelmert. 

 Leipzig, iSSo, 18S4. 



