56 GEODESIC KSTVESTIGATIOIn'S. 



But having V and a close approximate to I" we can find the 

 geocentric radius r' to 8' and a close approximate to r" the 

 geocentric radius to S"; and therefore, since we may consider the 

 comparatively short chord c as known accurately, we can find the 

 angle which this chord subtends at the centre of the earth with 

 an extreme degree of approximative accuracy. Hence, in the 

 spherical triangle we may consider the sides s'p, s's", and the 

 angle between them as known, and obtain by Napier's Analogies 

 the remaining parts, viz. : — The angle ps"s' which is the geo-^ 

 centric azimuth A^^ of the station S' as taken from S", — the 

 angle s'ps" which is the difference of longitude o of the stations 

 S', S'', — and the side ps" which is the geocentric latitude \" of 

 the station S". We can also obtain, by a well-known formula, 

 the geographic latitude I" of the station S" from its geocentric 

 latitude ^", and therefore the angle y" of the vertical at S" which 

 is equal to I" — \". 



Now, having the angle v" of the vertical at S", the geocentric 

 azimuth A^^ of the station S' as if taken from S", and the circular 

 measure 2 of the geodesic arc d, it is evident, from an observa- 

 tion already made, that we can find i", A'' and a": — 



For if from *S" we let fall a perpendicular on the tangent plane 

 at the station >S'", and that we conceive a sphere described having 

 S" as centre and the distance to the foot of the perpendicular as 

 radius, and that i'', e",f'' are the points in which it is pierced by 

 the chord c, the normal S"J^", and radius CS" of the earth ; then 

 it is evident we have on its surface a spherical triangle e'f'i" of 

 kindred kind to the triangle ^fV on the sphere having >S" as 

 centre, and that one side is 90° +42, another side "", and the 

 angle opposite the first side supplement to A,^. Moreover, the 

 three remaining parts are evidently the angle i" opposite side «'", 

 the angle A' between the sides v" and 90 + | 2, and the side a" 

 opposite the angle A' . 



