153 GEODESIC liS^TESTIGATIOXS. 



And since a!— a" = ^' + H — {A!'—Cl) — A!— A!' -f 2n 



.-. tan ( \ (A'-A") + n ] = ''"" ! Z~V, cot ^ a, (16) 



(. ; cosi (t +i ) 



©Multiplying both sides of equations (5) by the cliord k, and- remem- 

 bering that the projection 7c^, of the chord on the plane of the equator 

 is = & cos 6, we have 



Ic. sin A' cos J 2' =. k^ . sin ^, ") ■ ,-,h\ 



7c. sin A" cos i 2" = K"^ . sin (p,,) '" '" '" ^ ' 



But from the plane trianglej^^Cj;,,, we have 



, E, cos I' sin CO a,, cos I" sin o) 



sm (p^, sin (^, 



i • sin A' cos i 2' = R„ cos Z" sin a; ") /-i q\ 



^ • sin A" cos i 2" = S, cos t' sin coj 



A sin A' cos I 2 ' := r,, cos A" sin « ") /■• gv 



^ sin ^" cos i 2 " = r, cos A' sinwj ■" ■" ■" ^ 



25 sin i 2 



Again k ■='. 



■« 1 -e/ 



= i2„ cos Z" sin CO 



2 

 2*. sin 4' sin 1 2 cos ^ 2' 



2 



2.. sin ^- sin I 2 cos i 2- ^ ^, ^^^ .^. ^j^ ^_ 

 2 

 And if we assume cos I 2 = cos i 2' = cos i 2", these may be written 

 s . sin ul' . sin 2 = B,,, cos I" sin co "^ 



72~ C (20) 



8 . sin A!' . sin 2 =: J?, cos V sin co \ 

 2 -^ 



in which 5 and 2 are the length and circular measure of the geodesic arc. 



© 



(22) 



From the triangles D, S^ I, D^^ S^, I, we have — 



cos i 2' . sin A' = cos {z,, — i 2") sin D,\ ^o, v 



cos i 2" . sin A" z= cos {z—i 2') sin Z)^ J ^'^''^ 



And from the triangles S^PI, S,^FI, we have — 



cos I' sin CO = sin z^ sin D„ '^ 

 cos Z" sin CO = sin z„ sin ZJ^) '" 



From these we easily deduce — 



, sin A" . cos i 2" , , -/ /■„„-. 



cot ^^ =^ : = -, — tan t 2 (23) 



cos I . sm CO . cos * 2 



. sin A' . cos i 2' , , _,,/ /■„ .v 



cot 2,/ = -. = — tan I 2 (24) 



cos I . sm CO • cos i 2 



or, with close approximate accuracy for mutually visible stations — 



cot z, = ^'" ^" — tan i S! (25) 



(26) 



