12 On Practical Geodesy. 



And should the latitudes of the stations be equal, then 

 putting I for the common value, we have the rigorous 

 formula 



tan J C = sin I ' tan J w 



or, since the tangents of small angles are proportional to the 

 numbers of seconds in the angles, we have, approximately — 



C" = sin ^ • 0)" 

 in which C" and w" represent the seconds in the "conver- 

 gence " of meridians, and in the difference of the longitude 

 of the stations. 



16. And applying Todhunter's formula pertaining to 

 spherical excess (see page 72, formula 3, of his trigonometry) 

 to the same spherical triangle, we at once obtain the useful 

 relations — 



cotiz--cotir = -^"^t(^ + f--") 



2 2 COS J (A, + A,, + (o) , , 



tanir-tanir ^ _ cos HA. + A, + o.) 

 ^ ^ COS i (A, + A,, — (o) 



It is evident that instead of J l' and J r, we may write 

 (45° — 1 i;) and (45° — J l^;) in formula (49). 



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



. . sin A, cos a, . , sin A,, cos a,, 



sm d), = r^— t; : sm d>,, = ^^^— 'i 



^' sm 6 ' ^" sin 6 



sin A, _ sin <^, , cos a,, 



sin A,, sin ^^^ cos a. 



But from the plane triangle p.C^p,,, we have — 



sin <^. _ K,^ cos I,, 



sin <^,, E, cos I, 



.'. also the rigorous formula — 



sin A, E,,, cos I,, cos a,, , . 



sm A,, K, cos I, cos a, ^ 



And since for any pair of mutually visible stations, such as 



occur in trigonometrical survejdng, we may assume — ^= 1, 



.*. we have — - 



(-) 





sin 



A, 



K„ cos 



^. 









sin 



A, 

 cos I,, 



R, cos 



I. 







sm 



\/l- 



e' 



sin^ 



I. 



sin 



A, 



COS Z, 



¥ 1- 



e' 



sin^ 



l. 



sin^ A, 



_(1- 



- e') tan^ 



■I, 



+ 



1 



sin^ A,, (1 — e2) tan^ I, + 1 

 (true to at least 8 decimals places in theh logs.) 



