GEODESIC INVESTIGATIONS. 151 



Tlie arc Pi" is tlie measure of tbe angle whicli the chord 

 S'S" makes with the polar axis ; and is also the complement of 

 the angle e which the chord S'S" makes at i with the plane of the 

 equator. 



Let D^, D^^, be the points in which the great circles IS\,, IS^, 

 cut the great circles PS^C^, JPS^^G,^, respectively. 



The arc S^S^^ is the measure of the angle between the normals 

 S'Z^, S"Z^^; the arc Cfi^^ is the measure of the angle G between 

 the central radii to the stations ; the arc S,D^^ is the measure of 

 the plane angle S'Z^S" ; and the arc S^^D^ is the measure of the 

 plane angle S'Z^^S". 



The arcs S^C^, S^fi,^, are the measures of " the angles of the 

 vertical" at the respective stations S', S" . 



The arc 10 ^ and the supplement of the arc J(7„ are the measures 

 of the two interior angles of the triangle formed by the chord of 

 the geodesic arc, and central radii to its extremities. 



The small spherical angle S, IS^^ is evidently equivalent to the 

 dihedral angle A between the two normal-chordal planes. 



^^^ The interpretation of the points, lines, angles, &c., of the 

 whole figure can present no difficulty to any one, and need not be 

 further elucidated. But, in order to comprehend the investiga- 

 tions it is necessary to master the following notation, which is 

 adhered to throughout as closely as possible : — 



I', I" represent tlie geographic latitudes of the stations >S", 8". 

 i/! h/ Ji )> » colatitucTes „ „ „ 



a', A." „ ,, geocentric latitudes „ „ „ 



A', A" „ ,, geographic azimuths as ah-eady explained. 



2, represents the circular measure of the arc between S' and S" . 

 Tc „ ,, chord of the arc. 



5 „ ,, length of the arc. 



Tc^ „ „ projection ip,i^„ of the chord h. 



„ „ angle which the chord k makes with /rr^. 



(j)„ (p^, represent the angles Cp,]),„ C2J„p„ in the plane of the equator. 

 j8^, yS^, „ „ angles which the chordal plane parallel to the polar 



axis makes with the normal-chordal j^lanes S', S". 

 A represents the angle between the two normal-chordal planes. 

 O, „ either of the angles S^S^D^, S,,S^D,,. 



the arc 'S'^'S',,, or angle between the normals at the stations. 



the arcs I'D,, FD„. 



the arcs S^D,,, S„I>,. 



the angles FD,S,„ FD„8,. 



the lengths of the lines S'Z„ 8"Z„. 



the lengths of the central radii CS', CS". 



the lengths of the equatorial and polar radii. 



the earths eccentricity. 



the depression of the chord below the tangent plane at S'. 



)> >) )! )) " • 



the arcs S,D,, S,,T),,. 



R„ E, 



a, I, 

 e 



5/, 5„, 



