96 



Reduction of the formula for refraction to one convenient for 

 computation. — Comparison with Laplace's formula. 



9. From the equation which takes place, supposing the 

 surface of the earth a plane. 

 Viz. m sin. 6 = sin. C^'+Q) 

 We obtain, making m constant, 

 m sin. ^ = Q cos. (^-HQ) 



= Q cos. (^ + Q)— Q ' sin; (^+Q) 

 Hence making Q = and then w = m — 1 we have by 

 Taylor's theorem 



Q = (m— 1) tan. 6 + JH^JH- tan. ^ ^ + &c. 



taking m— 1 = ,0003 and 6 = 80'. 45' 



(in — 1) ' tan. ^ 9 n't 1 



^Tin. 1" ' 



the following terms are therefore insensible. 

 Hence substituting in equat. (7) art. 4. 

 We obtain for all values of 6 less than about 80°.45' 



R == (>»— 1) t<^n. fl {m—\)ltan. Q , 5 (m—\) I ' tan. ^ 6 



Im. V a COS.'' 9 sin. 1'' 2 a- cos. ^ 9 sin, 1" 



(ni— \) ' tan. 3 ^ /-JN 



+ 2«"». 1'' 



The two last terms are insensible except when is 



nearly 80'. 



10. The formula of Laplace (p. 268. tom. 4. Mec. celest.) 



I 



in seconds of a degree = -^i—[^i A + ——g > 



