182 GEODESIC INVESTIGATIONS. 



upon the axis of revolution, and e the angle which the geodesic makes with 

 the trace t^, — 



sm £=•£-'■ 



2°. If the traces of meridian planes cut a geodesic at right angles, then will 

 the perpendiculars from the points of intersection upon the axis of revolution 

 be equal to each other ; and the distances of the points of intersection from 

 the poles will be equals. 



3°. If the trace of one meridian cut a geodesic on a symmetrical surface of 

 revolution (such as the earth's surface) in two points perpendicularly, then 

 will the line joining the points of intersection pass through the centre of the 

 surface. 



4°. If i be the point of intersection of two geodesies on any surface of 

 revolution, and jS^, /S^,, the points on which they are cut perpendicularly by 

 traces of meridian planes ; then, if ^^, jj^,, g denote the lengths of perpen- 

 diculars from ;3^,'(S//, i upon the axis, and that i// represerits the angle between 

 the geodesies at their point of intersection — 



i|. = sin-^ {Ok 4.siu-i 0L\ , 



5°. If two geodesies cut each other in two points equidistant from the axis 

 on a symmetrical surface of revolution : then will their angles of intersection 

 at one point be equal to their angles of intersection at the other point. 



©From formulae (21) it is evident that when A' = A" , we must have 

 c„ = s^ = ^ 2' -f a 2". It is also evident that when sin A! =^ sin D^, 

 and that A' is acute, ancl D, obtuse, then will s^, := J 2' -}- i 2". And putting 

 7^ to represent the value of the angle A', in this case, it is evident that when 

 A' is greater than V, then will z^, be less than ^ 2' -f ^ 2" ; and when A' is 

 less than 7" then will s„ be greater than i 2' -|- i 2". 



Hence we have the following theorem pertaining to the figure of the earth, 

 or to any surface of revolution whose normals cut the axis in points analogous 

 to the way in which the normals to the earth cut its axis. 



Theoeem. — If S" be any fixed point within any convex closed curve on the 

 surface of an oblate spheroid whose adjacent pole is P^, and that Z^, is the 

 point in which the normal to the .surface at S" cuts the axis ; there are 4 real 

 points S' on this curve, and 4 ^joints only, such that the angle S" Z^, S' sub- 

 tended at Z^^ is equal to the sum of the angles of depression of the chord 

 S" &)' below the tangent planes at S" and S' . 



1°. The two points in which the curve is cut by the plane iV through S" 



perpendicular to the axis. 

 2°. The two other points lying on the same side of the plane N, and such 

 that the azimuth of S' as taken at S" is acute, and the azimvith of 

 S" as taken at S' is also acute, but greater than the other, and 

 approaching very nearly to 90° when (as in case of the earth) the 

 ellipticity of the surface is small. 



It may perhaps be proper to mention that the circular measure of 

 the arc between the stations, which is given as data in some of the problems 

 of this paper, is supposed to be more accurate than is usually obtained from 

 the mere triaugulation calculations of a trig, sm-vey. In a future paper, when 

 considering three stations simultaneously, I will show that the circular 

 measures so obtained must, if used, lead to very loose determinations of 

 latitudes, azimuths, and diiFerences of longitude. 



Sydney : Thomas Eichards, Governmeut Printer.— 187 1. 



