S^ On Practical Geodesy. 



And putting M to represent the reciprocal of the dexter of 

 this equation, we easily find — 



sin^ ln= -% — (-2 — sin^ l\ • M^ (i 2 4) 



an equation expressing the latitude of the n^^ station in 

 terms of the latitude of the 1st station and the sines of 

 the angles of depression of the n — 1 chords joining the 

 consecutive stations. 



2. We have also the rigorously accurate relations 



R2 COS I2 _ sin Aj 2 cos a, ^ ^ 



Rg COS Zg _ sin A23 cos a, 3 



K, cos I, sin Ag^ cos a^^ ' 



R3 cos l^ sin A3 2 cos a^ ^ 



and .'. 



TR, nop / siri A Rin A .,., 



cos a cos a / 





• 1 2 ^^° "'2 3 /, „ 



K. cos I sin A sin A .... 



cos a cos a 





*^UD Ug J l.UOU,3 2 



J{l-e')t2.nH^ 



+ 1 



J {I- e') tan^ 4+1 



and from this we easily find — 



„ / 1 \ /sin A„ , • sin A 



tan^ 4 = ^^tan^ l^ + j-3772; ' \ 



3 2 



sm A^, .sin A, 3, 



/ cos g^^ • cos gg^ V ]_ 



Vcos g, • cos g„ , ' 1 — 



an equation expressing the latitude of the n'^ station in 

 terms of the latitude of the 1'* station, the azimuths, and the 

 angles of depression of the chords connecting the stations. 

 3. And from (123) and (125) we have- 

 cos l^ 



cos I. ^^„ , K.XXX -.^ 



(127) 



sin 



A,, 



• sin 



A,,.. 









sm 



A., 

 tan 



• sin 



A3... 

 tan g 











32 



tan gj 2 



4. Let 1, 2, 3, n — 1, n,he any odd number of 



stations on the earth's spheroidal surface, such that none 



of the chords (12), (23), (^ — 1, ^^), exceeds 100 



miles in length. Then, from formula 49, it is evident we 

 have the relations — 



tan (45° — i I,) 

 tan (45^ — i Q 



cos 



i 



(A, 3 



+ A3, 



+ <",3) 





COS 



1 



(A,3 

 cos 



+ A3, 



J (A3 3 



+ A3, + 



",.s) 





cos 



i(A3 3 



+ A3,- 



<■>..) 



