164 GEODESIC INYESTiaATIOXS. 



Problem 2- 



Griveu the azimutlis A', A", taken at the stations S', S", and the lati- 

 tude I' of one of the stations <S" : to find — the latitude I" of the station S" ; 

 the difference of longitude oo of the stations ; the circular measure 2, the 

 chord k, and the length s of the geodesic arc between the stations ; the angle 

 1/ -which the two normals make with each other ; the angle A between the two 

 normal-chordal planes ; &c 



To find I", we have 



, o y// sin- A" f, 07/1 «^ s o- 



tan- I ^= — :^ (tan- i -4- — ) — — 



If we use the ratio of the squares of the equatorial and polar radii of 

 the earth as determined by Bessel, the above formula can be written : — 



tan^ I" = ""!, ^", (tan- I' -f 1-00671945) — 1-00671945 

 sm- A 



To find the difference of longitude w, we have — 



tan \ CO = '°' : II 7 12 . cot ^ W + ^") 

 sm (t -j- ") 



To find the circular measiu'e 2, chord 7c, and length of arc s, we 

 have 



, 1 _ cos r cos A' 4- cos I" cos A" ,_ _ i 

 ^^" * ^ == ■ ^.sinr+i^..sinr • (^'^")^ 



■, S, cos I' sin a> 



sin A" cos ^ 2 

 7c -2 . 



2 sin i 2 



We can find the circular measure otherwise, by first finding the angle 

 n which a plane parallel to the two normals makes with either of the normal- 

 chordal planes, and also the angle v between the normals. Thus : — 



From the triangle S, PS^, we have formula (16) — 



tan f i {A'~A") + n] = '"'tlf/T? - cot i a> 

 I ^ ^ i cos i (i' 4- 1") 



from which to find H. 



^ . sin ir (V -f I") . . 



* cos i 1'= = ■ —' . Bin t a» 



cos i {A' -\- A") 



from vrhich to find v. 



Then the following set — 



tan i 2 = tan J y. cos n . 



T Sy cos I', sin CO i2^/ cos I", sin co 



sin A", cos i 2 sin A', cos i 2 



^. 2 



s = 



2. sin i 2 



tan ^ A = sin I 2. tan fl 



sin ^ A = sin J v. sin Xl 



from which to find 2, Ir, s, and A. 



