150 GEODESIC INVESTIGATIONS. 



Let Cp Sj, I, P, be the points iu wliicTi the sphere ;S' is pierced 

 by the productions of the lines GS', Z^S', S"S' through the centre 

 >S", and by the line aS"P parallel to the polar radius and in the 

 same direction. 



Let C\^, S^^, I^^, Pi, be the points in which the sphere 8" is 

 pierced by the productions of the lines C>S", Z^^^" , by the chord 

 B^8" taken in the direction /^''aS", and by the line S^^P ^^ parallel to 

 and in the same direction as the polar radius GP^. 



Then, evidently, the points P^, C;, /S'^ are in the trace, on the 

 unit sphere ;8", of the earth's meridian plane through /S"; and the 

 points P II, Gil, 8 II, are in the trace, on the unit sphere 8" , of the 

 earth's meridian plane through the station 8" . 



We have, evidently, the arc PI := the arc P nl,! ; arc 

 Bil— 90°— i 2'; arc njn = 90° + ^ 2"; angle PSJ= supple- 

 ment of the geographic azimuth of the station 8" as observed at 

 S' (which azimuth we shall denote by the letter A') ; angle 

 Pii^ilzn the geographic azimuth of the station S' as if taken at 

 8" (which we shall denote by A!') ; arcs P8i, P^Sn equals to the 

 geographic co-latitudes of the stations *S", *S"'; arcs PG,, PGn, 

 equals to the geocentric co-latitudes of the stations 8', 8'' ; angle 

 PC^Zim supplement of angle G^, which we shall call the geocentric 

 azimuth of 8'' as if taken at 8' (this angle cannot be directly 

 measured with the iDstrumeuts) ; and the angle PuGnIn is the 

 geocentric azimuth Gn, which corresponds to the geographic 

 azimuth A''. 



^g^ In the figure, and all through this paper, the latitude of 

 the station 8' is considered greater, or not less, than the latitude 

 of the station 8". 



Now conceive the sphere 8" moved by direct translation along 

 the chord 8" 8', carrying its lines and points rigidly fixed, until its 

 centre coincides with the centre >S" of the sphere 8'. Then it is 

 evident the points In, Pn, will coincide with I, P, and that the 

 points I, Gi, Gil, will lie in one great circle of the sphere Sf. 



It is also evident that the points P811G11 lie in the trace of one 

 plane, or in one great circle, and that the spherical angle 81PS11 

 or GiPGi represents the difference of longitude of the stations 

 8', 8". 



Let Pi, Pn, be the points in which the lines P8', P 1,8" , pierce the 

 earth's equator. It is evident that the plane angle ji»^C;£»^^ is r= the 

 difference of longitude of the stations 8' , 8" . 



Let <?>/, <^„, represent the angles Gp^p^, Gp^p^, of the plane 

 triangle ji^^C^?^,. It is evident ^,, <p,„ are equivalents to the dihedral 

 angles which the chordal plane 8'8''piPii parallel to the polar axis, 

 makes with the meridian planes passing through the stations 

 8', 8", respectively. 



It is also evident that the spherical angle 8iPIzz: cpi, and that 

 the spherical angle >S^^PJ nr 180° — <p„ziz<!>, + the difference of 

 longitude of the stations 8', 8". 



