391' Mr. Ivory on the Theory of the Astronomical Refractions. 



d.^d = sm$x»{l+ct) x f* ^ -'^ . X 



»/ V cos* a + 2 IX 



This expression being regular, it may be continued to any 

 number of terms, and it has the advantage of being linear 

 with respect to the coefficients. Adverting to what x stands 

 for, it will appear that L x .r is nearly equal to 5, or to z^ that 

 is, to the elevation in the atmosphere ; so that, if we suppose 

 the greatest height of the atmosphere is 10 x L, or about fifty 

 miles, the greatest value of j; will be 10; and all the integrals 

 in the foregoing expression must be taken between the limits 

 zero and 10. But the quantity c-^ is so small when x has in- 

 creased to 8 or 10, that the results are not sensibly different 

 whether the integrals be extended to those limits or be con- 

 tinued to infinity. By substituting the values of the func- 

 tions, the expression of S 9 will take this form : 



/■ /-i d X c — ^ 

 8d = sinflxa(l+«) X -( j—, — _^ — t- 

 ^ ' \J V cos^ d + 2 2 jr 



J */ cos^ & + 2i X 



_ /• /* ^ .Uc-^'' — 3c-<'-\-xc-*) 



''J >i/cos^9 + 2?a; ' 



. fl r. — ^ .f8c-2*-8c-*-!- 1 XC-* 



''J \/cos«fl + 22a; ^ 



6 ' 

 ^ .(16r-^-16c2-*+16:rc-* 



-/'/ 



Vcos^Q + 2ix 

 15 , 



— x'c-'' 



,11 5 . x^c—''^ 



In order to illustrate the rapidity with which the terms de- 

 crease, it may be proper to find the limit of 8 9, by making 

 cos^ 6 = 0, and integrating between the limits x — 0,x = oo; 

 which limit is not sensibly different from the refraction at the 

 horizon. Now it will be found that, in the circumstances 

 mentioned, 



xfl _ «(i +«) v't r ( —^ \ 



