On Practical Geodesy. 3 



it is evident that the plane angle p^C^p^^ is equivalent to the 

 difference of longitude of the stations. 



It is also evident that the plane angles C^p/p^^, O^'pjp^, are 

 equals respectively to the spherical angle S^PI, and the sup- 

 plement of the spherical angle S^^PI. 



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

 IS^, cut the great circles PS^C^, PS/zC',,, respectively. It is 

 evident the arc S^S^^ is the measure of the angle which the 

 normals make with each other. 



The arc SD„ is the measure of the plane angle SoZ„S„„ ; 

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

 arcs S^C^, ^yf^.p ^^^ ^he measures of " the angles of the 

 vertical" at the stations SoS^^ ; the spherical angle SJS^^ is 

 equal to the angle between the two normal-chordal planes. 



And if 0, E^, E^^, be the points in which the great circle 

 of the unit sphere having I as pole cuts the arcs S^S^^, ^J^„, 

 S^^D^, respectively ; it is evident that the arcs S^E^, S^^E,^ are 

 the measures of the angles of depression of the geodesic 

 chord SoSoo below the tangent planes to the spheroidal 

 earth at the respective stations SoSo^; and they are the 

 complements of the angles which the normals make with 

 the chord. 



The spherical angles S^^S^D^^, ^^S^/I^^, are equivalents to the 

 angles which any plane parallel to the two normals makes 

 with the two normal-chordal planes. 



And the spherical angles ^J)J)^, S^D^^D^, are equivalents 

 to the angles which any plane parallel to the two lines 

 SqZoo, SooZq, makes with the normal-chordal planes. 



The interpretation of the other points, lines, angles, and 

 planes of the figure can present no difficulty, and no further 

 elucidation is necessary here ; but in order to avoid miscon- 

 ceptions, it should be remembered that all through this 

 paper (when two stations only are considered) we will 

 consider the latitude of the station So greater or not less 

 than the latitude of the station Soo, — as indicated in the 

 figure. 



NOTATION. 



Z^, 1^1 denote the latitudes of the stations S^, S^^j respectively. 



V, I" „ colatitudes, or the arcs PS„ PS,; „ 



L', L" ., arcs PD,, PD,. 



azimuths or angles PS,D,„ PS„D,. 



A,, A, 



A,, A, 



angles PS^S,, PS,S,, of the triangle S,PS, 



„ PD,S,,, PD,S,. 

 arcs S,D,, S,D,. 



