On Practical Geodesy. 29 



35. We will now consider how the magnitude of the 

 angle a varies when the stations S^, S^^, are supposed to be 

 situated on two fixed parallels of latitude, and at such dis- 

 tances asunder as may or can occur in trigonometrical 

 surveying. 



From equation 100 we^at once perceive thac when the 

 latitudes l^, l^^, are constants, the angle a between the 

 normal- chordal planes increases or decreases according as 

 cos^ I,, sin^ A,, — cos^ I, sin^ A^ increases or decreases. 



Or, if in this we substitute for sin^ A^ its equivalent as given 

 by equation 50, then we know that a increases or decreases 

 according as the expression 



sin^ A,, ( R^, cos^ I., — 'B?,, cos^ I, ' —^—^') increases or decreases. 

 ^ cos- a/ 



Now A^^ being the necessarily acute and lesser azimuth, we 



know that sin^ A^^ increases as the azimuth A^^ increases- 



And, since = ^ is constant, and that a^, and a 



' sin a, R,, " ' 



increase or decrease according as the difference of longitude 



1 -J.' -J XJ.T- ^ 1 — sin^a,, cos^ a,, 



0) increases or decreases, it is evident that :; . ^ or — 5—' 



1 — sin'' a^ cos- a, 



decreases according as the difference of longitude increases ; 

 and .'. that a increases as w and A^^ increase up to that point 

 at which the trace of the normal-chordal plane containing 

 R^^ touches the parallel of latitude on which S^ is situated. 



36. Other new and useful formulae can be easily derived 

 from the figure. For instance, from the spherical triangles 

 S,PI, S„PI, 



cos = sin a^ sin I, — cos a, cos l^ cos A, . 



cos — — sin a„ sin I,, + cos a,, cos I,, cos A,^ \^^'^) 



.'. sin a, sin l^ + sin a,, sin l,^ = cos a, cos l^ cos A, , . 



+ cos a,^ cos l,^ cos A,, ^ ^ ^ "-^ 



and hence with close approximation to absolute accuracy, we 

 have 



tan a, sin I, + tan a,, sin l,^ = cos I, cos A, -f cos l,^ cos A,, 



but tan a^ _ cos I, sin A, 



tan a„ cos I,, sin A,^ 



And from these we easily find 



cos l^ cos A, -f cos I,, cos A,^ 



tan a, = -. — -. — j — -. — 7 — ; -, — -. — -, — -. — 7— • cos L sin A, 



' cos I, sin C, sm A,+ cos I,, sin L, sm A,, ' ' 



II /I n II n , . 



COS I, cos A, 4" COS I,, cos A,^ . 



tan a,, = j — -. — -. — -. — r — ; -, — -. — -. — -. — r— • COS L, sm A,, 



" COS I J sin t, sm A^ -f- cos I,, sin t^, sm A,^ " " 



