54 GEODESIC I]!rVESTIGATIONS. 



Before proceeding to tlie question, I consider it necessary (in 

 order to render the investigation satisfactory throughout) to 

 state the following preliminary theorems, for the proof of which 

 the appended notes can be consulted. 



1°. If «i, ^2, . . . . «n he any number n of small and con- 

 secutively connected arcs forming one " geodesic" or 

 shortest arc d between any two points on the spheroidal 

 surface of the earth, and that p, p, . . . . p, p> are 



1 2 n n+1 



the lengths of the radii of curvature taken in order at the first 

 extremities of the series of arcs and at the final extremity 

 of the last of the series ; then will the circular measure 

 of the whole geodesic arc d be equal to the sum 2 of the 

 n terms of the series. 



2ai 2^2 2an 



P+P ' P + P P + P 



12 3 3 n n+1 



And the radius jffi = — is the mean radius of curvature of d. 



2°. If a geodesic arc d connecting any two stations S', 8" 

 on the spheroidal surface of the earth be not more than 

 60 miles in length, and that p', p" are the radii of curva- 

 ture of the arc at the points S' , S" respectively ; then, if 

 on the normal at either station we assume a centre whose 

 distance from the station is equal to p'+p" , the circle de- 

 scribed from such centre with p'+p" as radius will pass 

 through both stations S', S". " 



And , , „ will be the circular measure of the angle between 



P+P . . '^ . . 



the chord connecting the stations and the straight line 



drawn from either station, in its tangent plane, to the 



foot of the perpendicular from the other station on such 



tangent plane. 



3°. Having the length of a geodesic arc d on the spheroidal 



surface of the earth, and the geographic latitude I' of 



one extremity, and also the geographic azimuth of the 



other extremity as observed from the first, — there are 



simple, well-known methods of computing the latitude 



I" of the second extremity to within one or two seconds 



of accuracy, and the azimuth of the first as taken from 



the second to within one second of accuracy. 



Let C be the centre of the earth, PQ its polar axis, PS'Q, 



PS"Q the geodetic meridians through the stations S', S", and let 



JE' be the point of intersection of the normal at S' with the polar 



axis. Let 771 be the point in which the line S" m parallel to the 



normal S'JS' pierces the tangent plane to the spheroidal surface 



of the earth at >S" (the plane of the horizon at S'). 



