GEODESIC INVESTIC4ATI0*]SrS. 55 



Conceive a sphere described having ;S" as centre and >S" on' as 

 radius ; and let i! be the point in which the chord c connecting the 

 stations pierces the surface of this sphere. Let e' f be the 

 points in which its surface is pierced by the productions of the 

 lines H' S', CS' through ;S" ; and let *S'' n' be the trace of the 

 meridian plane PS'S' upon the tangent plane to the earth at ;S". 



It is evident that — the arcs e'm', e'n' are quadrants ; that arc 

 n'm', or its eqviivalent, the spherical angle n'e'm' is equal to the 

 given geographic azimuth A' ; that the arc m't' is equivalent to 

 the angle between the lines S'S", S'm', or to half the circular 

 measure 2 of the arc S'S" ; that the arc e'f is the angle 

 v' of the vertical at *S" ; and that the angle e'/'i' of the spherical 

 triangle e'fi' is equivalent to the angle between the plgnes 



CS'/'F, CS'fi'S", or to the geocentric azimuth A. 



1 



Now we can easily compute the angle v' of the vertical at aS" 

 from the given latitude V of the station S' . 



And it is well known that by means of close approximate 

 values for I' , I", A', A", we can obtain extremely accurate values 

 of the circular measure 2 of the arc d and of the length of its 

 chord c. And therefore, in the spherical triangle e'fi' we may 

 consider as known : — the side ef (equal to the angle of the 

 vertical v' as ;S") , — the side e'i' (equal to 90°+ 2 ^), — and the 

 angle f'e'i' (equal 180" — A'). Hence, from this triangle we can 

 find, by means of Napier's Analogies, the angles e'fi',f'i'e^ which 

 are the respective values of A^ and i' ; we can also find the side 

 f'i' which is the measure of the angle between the geocentric 

 radius CS' and the chord c of the arc d. 



It may be proper to observe here, that if we were given the 

 geocentric azimuth A, instead of the geographic azimuth A', then 

 we could obtain A', provided we had the arc e'i', and arc e'f : for 

 in the spherical triangle we should have two sides and an angle 

 opposite one of them, from which to find the supplement of the 

 angle between the sides. In fact, a like case occurs when in the 

 sequel we have to obtain A" from a kindred spherical triangle, 

 and it is for this reason the observation has been made in order 

 that the process of investigation may be the more clearly com- 

 prehended. 



Now conceive a sphere concentric with the earth, and let 

 p, s', s", be the points in which it is pierced by the central vectors 

 CP, OS', CS". It is evident that s'^s" is a spherical triangle in 

 which the side s's' is equivalent to the angle that the geodesic 

 arc d or its chord e subtends at the centre of the earth ; it is 

 also evident that the side^.?' is equal to the geocentric co-latitude 

 of the station *S", and that the angle ps's" is equal to the geocen- 

 tric azimuth of the station ;S"' as if taken from the station S". 



