i 



Mr Sang on Optimum Surveying. 341 



meridian, and z the polar axis : putting also a for the equa- 

 torial, /3 for the polar radius, the equation of the spheriod is 



«*^ ^ «^ ^ /j2 

 hence, X Y Z being the co-ordinates of any station we have 



and for the equation of the horizon there 



(X-x) X (Y-y) X ^ ( Z-.) X ^ ^ 

 «« ^ /S'* "^ y^ "^ 



wherefore the equations of a vertical line at that station are 



^.(X-x) = ?l(Y-y)=^(Z-=). 



hence the latitude and longitude of the station A being known, 

 its rectangular co-ordinates are thus found. 



Putting (A) z= V{a.^ cos lat A- -f /3.2 sin lat A*} 



x^ ::::: ,— — . cos lat A . cos Ion A , 

 (A) 



^A = 7-z\ ■ cos lat A . sin Ion A , 

 ^ (A) 



z. — ,-rT sin lat A 

 A — (A) 



Leading along the vertical at A, a plane inclined at a given 

 angle AN (the bearing of N from A) to the meridian, the 

 equation of that traverse plane is 



(j'^^a;) {cos Ion A . sin lat A . sin A N + sin Ion A . cos A N J 



+ (^A — y) {s^^ ^®^ -^ • ^^^ ^^^ A . sin A N — cos Ion A . cos A N J 



— {Zj^—z) {cos lat A . sin A N} = O 



whence the value of a perpendicular let fall from N upon the 

 traverse plane intended to pass through it is 



-^ 5 cos lat N . sin lat A . sin A N . cos (Ion A — Ion N) 



+ cos lat N . cos A N . sin (Ion A — Ion N) [• 



— -^r X \ sin lat N . cos lat A . sin A N 

 N) \ 



_'!!z±! « sin 2 lat A. sin AN 

 (A) ^ 



This perpendicular let fall from N upon the traverse plane 

 A N, is exactly analogous to that used in the investigation for 



