On Practical Geodesy. IS 



From either of these we at once perceive that, with 

 respect to mutually visible stations, the ratio of the sines of 

 the azimutJis will remain sensibly constant when the lati- 

 tudes of the stations are of constant magnitudes, no matter 

 how the difference of longitude or the intervening geodesic 

 arc may vary in magnitude. 



18. If we find an angle V such that — 



, K,. cos I,, / V 



tano- = :^ f (54) 



K, cos Z, 



then from 51, we derive — 



tan I (A,- A) ^ , _ ^^ ^ ^ 



tan \ (A, + A,) ^ ' ^ ^ 



.-. tan \ (A, — AJ = tan \ (A, + A,,) • tan (o- — 45°) (se) 



tani(A,-A.)=£^i4^^i^ • tan (cr - 45°) • cot J <o (.,) 



From this equation it is evident that when the 

 latitudes are constants, then the greater w is, the less will 

 the difference of the azimuths be. We already know that, 

 in such case, the less also will be the sum of the azimuths, 

 and .'. the less will each of the azimuths be. 



19. It is evident that A^ — A^,, = A, — A,, + 2 O 

 and .*. 



and from this and (57) it is evident that when the latitudes 

 of the stations are constants in magnitude, we have 



tan{i(A, — A,,) + n} . . 



^-^ — ' ,^ "' ' , -* = constant. 



tan I (A, — A,,) 



And since the greater the difference of longitude of the 

 stations is, the less A, — A^^ must be ; .'. the greater w is, the 

 less will O be. 



20. From the spherical triangle S,PS,,, we have 



sin (A,, — O) _ sin V 

 sin (A, + O) ~ sin I' 



sin A,, sin I' — sin A^ sin V 



.*. tan O = 7 ; jr. — ; \ -. j. (59) 



cos A,, sin V + cos A, sin V ^ ' 



t^ In such cases as occur in trigonometrical surveying 



the angle O will range from zero to a limiting value of about 



10',, 00". In the case of the worked-out example in the 



sequel, the value of O is 7',^ 22" nearly. 



21. From the spherical triangles S,PI, S,,PI, we have — 



sin ^ sin <^, = sin A, cos a, 

 sin Q sin ^„ = sin A,, cos a,^ 



