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16 On Practical Geodesy. 



From the triangles S,S,,Z,, S,^S,Z,„, we have— 



k ' cos (z, — a,) 

 sm 2, = ^ ^' 



k ' cos (z,. — a J 

 sin z,, = ^ ^^ 



And for stations which do not differ in latitude by more than 

 1°, we know that cos (0 — a^ cos (z^^ — a^), and cos J ^, 

 are the same to 8 places of decimals in their logarithms; 

 .'. for such stations we have the closely approximate for- 

 mulae — 



k ' cos A 5 

 sin z^ = — -^ — 



^' (-) 



But in order to find z^ and 0^^ in the actual practice of 



trigonometrical surveying (the latitudes of the two stations 



being such as do not differ by more than 1°) we have the 



well-known simple formulse — 



_ s 



^' ~ R, • sin 1" , . 



(75) 



& 



Zn = 



R,, • sin 1" 



which enable us to find z^ and z^^ to within toVo part of a 

 second of rigorous accuracy. This can be easily seen from 

 the following — 



We have the rigorously true equation — 



R, • Q, • cos 8, = R,^ • Q,, . cos S,, 



in which (as is shewn in the sequel) 8 and S^^ are always 

 each less than 16 seconds, and differ from each other by less 

 than 0*2"; and as we know that under such circumstances 

 the logs, of cos 8 and cos 8^^ will be the same to 10 places of 

 decimals, .*. we can assume — 



R, • Q, = R, • Q, 

 But R/ -h Q/ = R,,2 _|_ Q 2 absolutely, 

 R^ = Q,, nearly 

 R,^ = Q^ nearly 



Hence if I^, \^, be put to represent the bases of the isosceles 

 triangles having the angles , ^, as vertical angles, and 

 sides equal to R^, R^^, respectively, we have — 

 1/ = r; + r/ _ 2 R/ cos z, 

 = R/ + Q,; — 2 R, • Q, cos s, 



