Geodesic Lines on the Earth's Surface, 357 



3. 



We now leave this part of the subject, and turn to the consi- 

 deration of the plane which, containing the normal at A, passes 

 through B. Let Q be any point in the curve of intersection of 

 this plane with the surface of the spheroid, co the longitude of Q, 

 U its reduced latitude, then 



sin U— cos U tan u 



2 f • n • cos U sin o) , . , N \ 



— e z < sin U - sin m, 1 —. , (sin u — sinw, >» 



L cosu'smco v "J 



We shall not stop to prove this equation (which is exact), but 

 merely indicate that it is most readily obtained by getting the 

 equation of a plane containing the normal at A and making a 

 given angle with the meridian at that point. In this equation, 

 substitute first the coordinates of P expressed in terms of u, a, 

 then those of B in terms of u 1 , co', and between the two equations 

 so obtained eliminate the azimuth angle. The result is the 

 equation just written down. Since in the right-hand member we 

 may neglect e 2 among the terms within the parenthesis, then by 

 the use of (5) and the following, which is easily proved by a 

 spherical triangle, 



sin a' sin U = sin a sin v! + sin cr t sin u t , 



we get 



. TT TT o . /sincr + sincr,— sin a'\ 



sin U — cos U tan w n = e z sin u. I : — -. 1, 



u ' \ sin cr / 



or 



TT e 2 4 sin A a sin i cr. . ,„. 



U=w n +— — -. — r^ — cosw n smw.. . . (7) 



u 2 cos -J & u 1 



It may be remarked that in this equation cos u may be re- 

 placed by cos m, since by (6) the difference is of the order e 2 . 

 We may apply this equation (7) to the plane which, containing 

 the normal at B, passes through A. If Q' be a point in the 

 curve of intersection, having the same longitude or on the same 

 meridian with Q, and if U' be its reduced latitude, 



tt; , e<2 4 sin i cr sin \ a. . . /Q . 



U'=w n + — — : — r* — *cosw sinw'. . . (b) 



u 2 cos f & 



4. 



We now know from (6), (7), (8) the relative positions in lati- 

 tude of the three points P, Q, Q', which are on any the same 

 meridian, P being on the geodesic, and Q Q' on the two plane 



