300 



SURVEY OP THE COAST 



tion of the two planes, d being any point observed on the 

 limb. Drawing the arc dc perpendicular to ac, the corre- 

 sponding point in the horizon will be c, and ac will be the 

 reduced arc. Calling the inclination of the planes dac=^ ; 

 the constant equal angles between the legs=^, (commonly 

 three;) and the distance of the point cJ from the intersection 

 of the two planes=?; the series for the reduction of the ec- 

 liptic to the equator will give the corrections for each suc- 

 cessive position as follows : 

 For the first position, 



ig' 



ad — flc=<r=-^-^,sin.2'?' . ^ 



For the second position, 



'■^'^°^,sin.4^-t-^%-sin.6», ^c. 



sin.3' 



ad'-~ac'=<r'=^^,sin.S(»+/8)— ^li^sin.4(?.t-^)-f- ^^,9in6 (,+/?), ^c. 



Z'.~ac'=<r'=-¥-^,sin.S(»+/8)- 



sin.2" 



sin.3' 



For the third position, 



ad" — ac"=t" = %i^,%v[\3(<^+n^)—0^,%\nA(*+2^), &c. 

 sin.l" ^ ^ sin. 3' ^ ^' ■' 



and so on for any greater number of legs and positions of 

 the instrument. The sum of any number of such correc- 

 tions being taken to ascertain the total correction as here- 

 tofore, and ordered according to their common factors, the 

 following expression will result : 



-M-^\ sin.3f-f-sin.S(*+/3)-fsin.g(*-f-2^)+, 5)C. 

 - ^ "", sin.4f+sin.4(»+/3)+sin.4("?-f-2y3)-}-, ^'c. 



t+i'+i'= J 



+^4^rsin.6»-fsin.6(»+/8)+sin.6Cf+2,s)+, kc. 1 

 sm.3'[_ ^ J 



and so in case of more legs. 



