162 GEODESIC INVESTiaATIONS. 



* The triangles S,PD", S"PDj, gire the following formiilse from which to 

 obtain the azimuths A', A", and angles B,,, D, ; — 



cos i (X„ — I) 

 tan i (J + A.)=cosi(i,, + Z,)" 



cot 



sin i (L,, — ?.) , 



tan H^' -!>.)=, inHi:^6° 



cos i (^, i.) , 



tanHA+-i")=^,^4-t 



cot 



sin ^ (?„ — ij 

 tan \ {D, — A )=:^ — ttt — r^^\' 



And since S^D, = i^ — 7, ; and 'S',/Z>„ = l„ — L,, ; we haye from the tri- 

 angles S,ID„ S,,ID,„ the following set from which to find IS,, ID,, IS,,, ID,,, 



sin i (D, + A' ) , , ,T ,x 

 tan i (7^, + 7D.) ^ ,i„ . (z^;,^. ) • ^^^ * (^'-^') 



cos* (A + ^4.' ) , 1/7- ^^ 



tan i {IS,— ID,) = -Y~^ — -T^y tan i (i — ?,) 



^ ' '^ cos i {D, — A J 



tan i (i«„+JO„)= ;i;{;j:+gj . tan i R,_i„) 



Then i2' = I TT — IS, ; ^2" = JS„ — t tt ; 

 z, = ID,, — J;?, ; z„ = 7<S'„ — ZZ), ; 

 from which to find ^-2', ^2", z„ z„. 

 •'. 2 = ^2' -|- 12" is also known. 

 To find the chord 7c and length s of the geodesic arc connecting the 

 stations, we have — 



T S, cos I' sin CO 



sin ^". cos ^2" 



jffi„ cos Z" sin £0 



I sin 4'. cos i2' 



2 sin 12 '' ' ' sin i (|2'-f-i2 



or we can find s at once without finding k, from — 

 S, cos I' sin o). 2 

 * "^ 2 sin A" sin i 2. cos i 2" 

 -K„ cos I" sin CO. 2 



2. sin A" sin •^2. cos ^2' 

 We can find the angle A from either of the triangles S,D,I, S„D„I, in 

 each of which we know the three sides and the remaining angles. 



We can also find the angle v (the arc S,S,^ between the normals, from 

 the triangles S,PS„, of wliich we know the two sides and included angle w. 



^^° This method of solution is rigorously accvirate as to the determination 

 of A', A", 1 2', i2", no matter how distant the stations may be from each other. 



And we mnst have — 



sin A' R„ cos I" cos ^2" 



sin A" R, cos V ' cos ^2 



* The direct expressions for the cotangents of the azimuths (as given in the Account of 

 the Trig. Survej' of Great Britain and Ireland) can be at once obtained from the triangles 

 PS, De, PS„Di, by applying the well known formula (Todhunter's Trigonometry, art. 44) 

 and using the above expressions for cot Lg, cot L^ 



