149 



GEODESIC INYESTIG/VTIONS. 



CoifTArN'iisrQ numerous new* theorems in Practical GreocTesy, and 

 solutions to all the problems relating to the determination 

 of latitudes, azimuths, difference of longitude, length and 

 circular measure of geodesic arc, &c. 



By MABTlif GrAEDiNER, C.J©., of the Qiieeii's University, Ireland, Member of 

 the Mathematical Society of London. 



\_Itead before the Eoyal Society, 5 November, 1873.] 



{See 2}lafe.) 



Let Pg be the pole of reference of the spheroidal earth ; C the 

 earth's centre ; S', S", any two stations on its surface ; Z^, Z^^, the 

 points in which the normals to the surface at *S", S", cut the polar 

 axis. 



The planes S'S"Zi, S"S'Z^^, shall be referred to as the "normal- 

 chordal" planes. The first of these planes, which contains the 

 normal to the surface at the point S', shall be designated the 

 normal-chordal plane ;S" ; and the other, which contains the normal 

 to the surface at the point S'', shall be designated the normal- 

 chordal plane *S". 



And any plane whatever which contains the chord of the geodesic 

 arc S'S" shall be referred to as a chordal plane. 



The polar and equatorial radii of the earth being something of 

 about 3,950 miles, and 3,963 miles in length, it is evident that 

 (owing to the great magnitude and small ellipticity of central and 

 normal sections) , for arcs of not more than 100 miles in length, 

 we may, with due respect to the most minute accuracy attainable 

 in actual calculations, assume that the normals to the surface at 

 the stations make angles with the chord connecting the stations 

 whose sines are equals. And we may consider the traces of 

 the two normal-chordal planes as equals in length and in circular 

 measure to the true " geodesic" or shortest arc between the 

 stations. 



Let s and 2 represent the common length, and common circular 

 measure of either of these plane traces on the spheroidal earth. 

 And let | 2', i 2", be the depressions of the chord of iS'S'' below 

 the tangent planes at S' and S". Conceive two imit spheres 

 described having >S", S" as centres. 



* All the theorems in the paper are given as netv, with the exception of (1) 

 and (18) ; and all the j)robIems, with exception of problems 1 and 5. 



