108 On the Difference of Longitude deduced from t^joo Stations. 



Let p and <p' denote the depressions of the chord y below 



the horizons of the first and second stations; then, according 



to what is shown at p. 242 of this Journal for October 1 828, 



we shall have these two equations : 



x' sin ui ■= y cos $ sin in 



X sin CO = y cos <t>' sin m' ; 



wherefore. sin m sin m' . , 



wjieieiuxe, __^ ^ COS <p = X COS A'. 



x' X 



By combining this equation with the former one, we readily 

 obtain, 



Sin [jx — m) cos ^ = sin (m'—ix!) cos i^'. 

 On the suppositions made; fj, — m and m' — jj.' are small arcs; 

 also cos <$) and cos (f.' are always nearly in a ratio of equality; 

 wherefore we may conclude without sensible error, that 

 Sin (ft— 7k) = sin (»«'— ju.'), 

 ?n + m' ^ iji, + [jj 



X — X' 



COS' 



Tin *" _ 2 m-lm' 



■^ . x+x' 2 



sin — 



2 



This demonstration comprehends the elliptical spheroid as a 

 particular case. 



When the two latitudes are equal, cos <p = cos (p' ; and we 

 learn from the equations (A') that ?« and ?«' are respectively 

 equal to one and another to [x, and /x'. But the formula for the 

 difference of longitude is true, independently of the situation 

 of the stations. 



In an elliptical spheroid when the latitudes are equal, the 

 excentricity disappears from the equations (A'), and it is there- 

 fore indeterminate. And when the latitudes are very nearly, 

 although not exactly, equal, there is so near an approach to 

 the condition which makes the excentricity indeterminate, that 

 no dependence can be practically placed on any result respect- 

 ing the figure of the eai'lh obtained by means of the equations, 

 or by means of the angles they conla"in. In order to find the 

 excentricity, we must have recourse to the measured distance 

 between the stations, as I have pointed in the last Number of 

 this Journal. 



If we put s for the geodetical line between the stations on 

 the spheroid, and represent by o- a line traced on the surface 

 of the sphere, in such a manner that every two jioints of 5 and 

 0- that are upon the same meridian, have the same latitude; 

 then the sum of the three angles of the triangle on the surface 

 of the spheroid, that is, the sum of m and the inclinations of 

 5 to the meridians at its extreme points, will exceed 180° by a 

 quantity proportional to the surface of the trilateral figure on 



the 



