Ixiv GEODESY. 



p'V sin ($ - Co) p"r sin 6 

 C — Co (in seconds) = r » 



p" = 206 2 64."8, log p" = 5.3144251. 



Attention must be paid to the signs of sin (0 — Cq) and sin 0, and to the fact 

 that angles are counted from A towards B through 360°. A diagram drawn in 

 accordance with the above specifications will elucidate any special case. 



b. Reduction of measured base to sea level. 

 Let /be the length of the bar, tape or other unit used in measuring the base. 

 Let /q be the corresponding length reduced to sea level for a height /i, this latter 

 being the observed height of /. Then if p denote the radius of curvature of the 

 earth's surface in the direction of the base, 



''=i^M-t+-y 



with sufficient accuracy. Hence, for the whole length of the base, 



2/0 = 2/ - - 2//?. 



If Z denote the total measured length, Zo the corresponding total sea level 

 length, and //the mean value of the heights //, the above equation gives 



p 



c. The three-point problem. 

 In this problem the positions of three points A, B, C, and hence the elements 

 of the triangle they form, are given together with the two angles ^/"C and BBC 

 at a point B whose position is required. Denote the angles and the sides of the 

 known triangle by A, B, C, and a, b, c, respectively. Also put 



ABC=p, BBC=a, 

 BAC = x, BBC = y. 



Then the sum of the angles in the quadrilateral BACB is 



a + ^-}-.r+;'+C=36o°, 

 whence 



K^ + ;■) = 180° - K« + /3 + c). (i) 



Compute an auxiliary angle z from the equation 



a sin ^ 

 tan z = 



Then 



tan 2 = , 



o sin a 



(2) 



tan \{x - y) = tan (z - 45°) tan ^(x + y). (3) 



These three equations give all the data essential to a complete determination 

 of the position of B. Any special case should be elucidated by a diagram drawn 

 in accordance with the specifications given above. 



