CHAP. IV.] RESULTS OBTAINED ASTRONOMICALLY 113 



probable error due to bearing as found in (6), expressed in 

 terms of seconds of diff. lat. and diff. long. 



III. Now calculate the mercatorial bearing and distance of 

 A B from the astronomical positions of A and B, not for- 



Fig. i8. 



getting to apply the spheroidal correction to the diff. long, 

 by meridian distance before calculating. 



We have already in II. (6) calculated the mean mercatorial 

 bearing and distance of A B from the triangulation. 



Project the two bearings and distances from a common point 

 A (Fig. 18), terminating in B and B' respectively ; then B B' 

 represents the difference between the two positions of B with 

 reference to A, as found astronomically and by triangulation. 



From B' drop perpendicular B' C on A B. 



Since BAB', the difference between the two mercatorial 

 bearings, is a small angle, B C may be considered as the 

 difference of the two determinations ofAB, orAB — AB'. 



B' C may be calculated from the formula — 



feet subtended x 34 , B' C x 34 



Angle in seconds = , or B A B = . 



distance in miles A B 



With B C and B' C in the right-angled triangle BOB' 

 calculate bearing and distance B B', and find the corresponding 

 diff. lat. and diff. long, between B and B'. 



Divide each of these in the proportion of the " maximum 

 probable error " of diff. lat. and diff. long, due to astronomical 

 positions and triangulation respectively, and the result will 

 be the amount required to apply to the astronomical positions 

 and to the triangulation positions to make them agree. 



8 



