MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 



207 



In order to illustrate the rapidity with which the terms decrease, it may be proper 

 to find the limit of § 6, by making cos^ ^ = 0, and integrating between the limits 

 X = 0, X =■ cv; which limit is not sensibly different from the refraction at the horizon. 

 Now it will be found that, in the circumstances mentioned, 



10 = 



V 9. 



X |l +x(^"2^- l) 



-/(2^/^-4) 



256 y' 



-/' (8^/21 ^'^'^ 



— &c. : 

 or, in seconds, 



S 6 = 2072"-46 -/' X 62-4 -/' X 10"-2 - &c. 



From this calculation it appears that the term multiplied byy" and all the subsequent 

 terms are too small to be sensible ; and as/' is much less than/, even the term mul- 

 tiplied by/ can hardly exceed a few seconds at low altitudes. There is great pro- 

 bability that the horizontal refraction is very near 34' 30", and does not exceed this 

 quantity. 



To prepare the foregoing expression of ^ ^ for integration, put 



m = 10, ;— = tan (p, e = tan -^ ; 



' cos a ri 2 ' 



then 



(1 — e^'^ 

 cos2 = -^^- X 2 i m, 



A: 



^/ cos2 ^ + 2 i m = ^^ • \/(l - e2)2 -|- 4 e2-|- = ^^' 

 and we shall have 



..=si„.x«-ii4f>x{r^4-'.o- 



For the sake of abridging, the several integrals in succession may be represented by 

 *io> Qi) Qj) Q3 ; so that the value of 5 ^ will be thus written : 



8 tf sin ^ X '-^^^ -(00 + ?^ Qi -/Q2 +/ Qs)- 



(C.) 



