Geodesic Lines on the Earth's Surface. 361 



7. 

 If in the expression for Su we make cr= cr l — \o J i we get for 

 8u at the middle point of the line, 



Su = 2e 2 cosu 7 — < —5 sin u— sin u, >. . (14) 



cos f a Ltan or J 



where u is the latitude of the middle point. We may apply this 

 to another example. Take the chain of triangles in India ex- 

 tending from the Kurrachee base in the west to the Calcutta 

 base in the east. The approximate latitudes of these positions, 

 taken from a common map of India, are 25° 0' and 22° 30' re- 

 spectively, and the difference of longitude 21° 10'. From these 

 we obtain 



«,= 92 58 



«'=101 34 

 a= 19 40 



u = 24 7 



and referring the geodesic to the curve formed by the vertical 

 plane at Calcutta, we get at the middle point 



tfo^ = 46-6 feet. 

 Also the differences of the azimuths of the geodesic from those 

 of the vertical planes are — at Kurrachee 2"'04, and at Calcutta 

 3''*76. The smallness of these quantities is the consequence of 

 the line being nearly perpendicular to the meridian. 



8. 



If we suppose the length of the line small, so that powers of 

 a' above the fourth may be neglected, (11) takes the form 



ou= — era, cos u < , (sin u — sm uf) 



+ ^(<r 2 + 3<r<7- / + o- / 2 )sintt y j>, . . . (15) 



by which the course of any geodesic of a few degrees in length 

 may be traced. If we may neglect a 4 , then comparing (15) 

 with (9) we find 



QP _ y + <r 1 , o— tr, 



QQ' ~ 3(7' 2 6a' ' 



showing that the geodesic lies between the plane curves. 



In the case of a geodesic joining two points on the same pa- 

 rallel, 



e 2 

 Su — -~ <T(T l (cr' 2 + So-a, + of) sin 2u. 



