Mr. Ivory's Solution of a Geodetkal Problem. 39 



sions of 8A, ju.— f*.', and <J>', into series containing the powers 

 otV. 



In two particular cases the general formulae become more 

 simple. 



First, when the geodetical line is perpendicular to the meri- 

 dian, the spherical triangle becomes right-angled, which ren- 

 ders the computation less complicated. 



Secondly, when the geodetical line is upon the meridian, 

 the angles of azimuth and the difference of longitude are 

 evanescent ; the difference of latitude is equal to s ; and the 

 problem is reduced to find s for a given value of s, from this 

 equation, viz. 



<ls = ds' v/'i+c*sin s (X+s'); 

 which, in almost every case, is sufficiently solved by the 

 formula, 



J l-j-e*sin«(X-|-£*) 



July 10, 1824. James Ivory. 



Postscript. — In the First Part of the Philosophical Trans- 

 actions for 1824, just published, there is a short paper by 

 Dr. Young, containing a rule for the astronomical refraction. 

 It is no more than a particular case of my general formula. 



If we write y for ~, and z for - ? , in the expressions at p. 422 



of the last Number of this Journal, we shall obtain 



m+l 

 ya(l_/)_«~ +t /y, 



f- 1 !!izi 



J 4 * m-l' 



With respect to these equations, it may be added, as al- 

 ready remarked at p. 450 Phil. Trans, for 1823, that when 

 m is less than 4, f is negative ; the total height of the atmo- 

 sphere is less than 25 miles ; and the rate of the gradation of 

 heat, although it agrees with nature at the earth's surface, in- 

 creases in ascending. . For these reasons I have excluded all 

 these cases in seeking the value of the exponent m, that will 

 best represent the real atmosphere of the earth; or, as Dr. 

 Young terms it, I have rejected them as inadmissible. 



Now put m = 2; then 



/= -ii 



y = $**-**■, 

 which is Dr. Young's assumption. 



In 



