80 Solution to a Case in Sailing. 



dividing by the modulus M =.43429, become 



A=7915.7 log. cotang. £(90°-L-^) 

 A'=7915.7 log. cotang. i(90°-L+£Z) 

 From these we obtain 



A-A'= P= 7915.7[log. cotang. £(90° -L-^)-log. cotang. 

 K90°-L + i/)]: 



P _ cotang.K90 o -L-£Q _ tang.j(90 o -L+£l ) 

 7915.7 ~ log 'cotang.i(90°-L+iO~ g "tang.i(yO°-L-^) ; 



P 



and if ^-nig - =log. n, we have 



tang.j (90°-L+^) _ cos.L+sin.j ^ 



n = tang.£(90°-L-^) = cos.L-sin.^ » 



w + l . 1 

 from which we find cos. L = -37 sin. %l, (2). 



To facilitate the computation by logarithms, put w=cotang. y, 



w+1 

 then i-=tang.(45°+?>), and equation (2) becomes 



Kb ~~~ i- 



cos.L=tang. (45 +<p)sin.^, (3), 



which with the equation 



P D cotang. C 

 log. cotang. <P=79l5^= 7915 . 7 . (*) 



furnishes a very easy solution to the question. 



In the example proposed we find by equations (1), C=32° 59', 

 1=208' '. The computation of equations (4) and (3) will be as 

 follows : 



L = 64° 08', . I. cos. 9.63969 



L+^=65°52', L-^=62°24'. The latitudes found by 

 middle-latitude sailing, are 65° 55', and 62° 27'. 



U. S. Naval Asylum, Philadelphia, August, 1843. 



