98 The Astronomer Royal on a Problem of Geodesy. 



the multiplication of small quantities we may consider 

 B = 90° — A and w'^=v^-\-iio\ 



The linear depression of B below the horizon of C is -y, 

 and therefore its angular depression is -^. That of C below 

 the horizon of B is sensibly the same. 



The linear depression of A below the horizon of C is ^^, 

 and therefore its angular depression is ^tt>. That of C below 

 the horizon of A is sensibly the same. 



The linear depression of B below the horizon of A is 



2V + 2W' 

 and therefore its angular depression is 



1/ v^ 110^ \_ / sin^A cos^A X 

 A2V'^2Wy~''\ 2V ■'" 2W /• 



That of A below the horizon of B is the same. 



Hence we obtain the following expressions for the three 

 reductions. 



For the right angle C, we must make 



E = 90°, cosecE=l, cotanE=0, D= ^y* ^'=2W* 

 The expression for the reduction becomes 

 <im __ a;^. sin A . cos A 

 4VW~ 4VW 

 For the angle A, 

 T^ w .2?. cos A T^, /sin^A , cos^A\ ■p_ a 



Substituting these quantities in the expression for the reduc- 

 tion, it becomes 



a: 



{sin A . cos'* A sin^A .cos A _ sin^A . cos A 1 



For the angle B, 



D: 



j7.sinA T^, /sin^A , cos^A 



D'-(i^+w)> E=«o°-A' 



2V 

 and the expression for reduction becomes 



gfcosA.sin^A cos-^A.sin A cos^A.sin A \ 

 ^l 8V2 "^ 4VW 8W2 /• 



