34 On Practical Geodesy. 



5. With respect to any three mutually visible stations 

 1, 2, 3, we can easily arrive at convenient expressions for 

 each of their latitudes in terms of their azimuths and differ- 

 ences of longitude. Thus — 



We have (49) and (128)— 



tan (45° — h h) ' tan (45° _ J Z.) = — CQ« \ (^12+ A,,+ <^n ) 

 ^ " '' ^ ^ ^ cos 1 (A12+ Kx— W12) 



tan (45°— I ^i) ^ cos \ (A13+ A314- (013) ^ cos \ (A32+ K^^ (O32) 

 tan (45°— \ 4) cos \ (A13+ A31— (Ojg) ' cos \ (A32+ A23— W32) 



tanM45°-iO -- '''? ft" t>^"t 



UC.U V ^ 2 ^ly cos 1 (A12 + A21 — (O12) 



, COS 1 (Ai3 + A31 + toi3) ^ COS \ (A23 4- A32 4- 0)23) 

 cos J (Ai3 + Agi — (Ok) ■ cos 1 (A23 + A32 — (O23) 



^ 2 2/ COS I (A 23 + A32 — CO23) 



, COS j- (A21 + A12 + (021) ^ COS i (A31 + Ai3 + o>3i) 

 COS J (A21 + A12 — CO21) ' COS 1 (A31 + Ai3 — W31) ^^^^' 



tan- (45° — \U = — cos \ (A31 + A,3 + <03i) 



cos J (Agi + A13 — CO31) 



^ COS \ ( A32 + A23 + ^032 ) _^ cos \ (A12 + A21 + <02l) 

 cos J (A32 + A23 — CO32) ■ cos \ (A13 + A21 W21) 



These equations are closely approximate to rigorous 

 accuracy, even when the stations are from 100 to 200 miles 

 asunder. 



6. Let Q, (^, Q^ be any three stations on the earth's 



spheroidal surface. Then if K^, K.^, Kg, indicate the angles 



between the chords joining the stations which have their 



vertices in (^, Q, Q, respectively; and that C^, C.,^ C3, 



indicate the corresponding angles of the geodesic triangle 



formed by the geodesic arcs connecting the stations; we 



have evidently 



cos K V 



cos C, = — tan a, „ • tan a, A 



^ cos a^3 cos a^2 1^ ^^\ 



cos K, \ / \ 



cos C, = = — tan a„ , * tan a„ „ / (13 2) 



2 cos a^ ^ cos ttg 3 21 2 3/ \ / 



r^ cos K 



cos (J, = — tan a„., ' tan a. 



cos a,„ cos a, , ^- 



