ELISHA MITCHELL SCIENTIFIC SOCIETY. 119 



where Xj ^ — x^ is the difference in the ordi nates <:ompiited 

 by the approximate formula above and s^ — s ^ length of 

 arc between them. 



We have just seen that this formula is true to the near- 

 est minute of arc when x^ and ^ are zero, in which case i 

 reduces to __, and we shall now reduce it and express 2 in 

 terms of X and X^ and test it for other values. 



We have, 



x' = }i as = % as' = % ^^ N^; 



a r^(X,^ — N^) ac' 



s, — s 7, ■ c (X, — X) 3 



(X, 2 + NX, 



Replacing ac- by its value (eq. 8) and multiplying 



1800 



I Sox 60 

 both sides of the equation by to reduce to minutes, 



we have, 



i (in minutes) =1 2 (Xj - -f Xj X — X^) (23). 



We find that this formula is correct to the nearest min- 

 ute, or as accurate as the formula a = ^/^ <7, w^hich is cor- 

 rect up to X = 14 (and practically to X' = 15) to the near- 

 est minute. 



In the above, Xj has been taken as the number of the 

 forward station and X^ of the one nearest S; but if the 

 reverse obtains, on interchanging X"j and X"" in the first 

 formula, we shall arrive at the same formula (23), which is 

 thus perfectly general and applies whichever is the forward 

 station. 



As an application compute 7-_^q : we have, putting Xj = 

 10, N = 5, 



V,o = 2 (10^' T 10 X 5 ^ 5-) =- 350' = 5°5o^ 

 correct by the general table, and in fact differs only a few 

 seconds from the exact value. 



