362 On the Course of Geodesic Lines on the Earth's Surface, 



If S be the point in which this geodesic cuts the meridian which 

 is equidistant from A and B, S' another point on this same me- 

 ridian such that a vertical plane at S' passes through both A and 

 B, and S" the intersection with this meridian of the plane con- 

 taining the normals at A and B, then, neglecting o 3 , 



S"S=f S"S'. 



The consideration of (15) or of (13) also makes it evident that, 

 under some circumstances or within certain limits of azimuth, 

 the geodesic joining AB will intersect one of the plane curves 

 between those points. 



9. 



Bessel, in one of his papers in the Astronomische Nachrichten, 

 gives, but without demonstration, an expression for the differ- 

 ence in length between the geodesic and one of the plane curves. 

 In case of the accident of a misprint, it is worth while to verify 

 it. Q being a point on the (southern) plane curve at a distance 

 aSu south of P on the geodesic, if we draw an arc of parallel 

 through P meeting the plane curve in the point R east of P, we 

 have PR = aSwtan a; the difference of longitude of R and P is 

 therefore 



Sco — Bu tan a sec u ; (16) 



and we may suppose all the points of the plane curve to be re- 

 ferred in this manner to the geodesic. Now, returning to the 

 expression for the length of a curve on the surface : when o> is 

 increased by hco, 



d£ \! d? + J 



is increased by 



2 dco fdhw\ L /dBa\ z r z d^ 

 ' ds\dz) + 2 \~aXJ ~dW' 



the first term of which, when integrated, is, by reason of the 

 character of the geodesic, zero. Hence the increment in length 

 ill passing from the geodesic to the plane curve is, since ds cos 



Bs = — J« 2 I cos 2 u cos 3 «( — m7 ) d%. 



As we require merely the first or principal term in the value 

 of hs and none else, we may put d£= — adu, and 



Ss = ^«cos 2 w / cos 3 a. A l-—j — \ du. 



