Analysis of the Mecanique Celeste of M. La Place. 255 



spheroid be one of revolution, but in the general case it has 

 a double curvature. 



The geodesic line is a curve whose first side is a tangent 

 in any given direction to the surface of the earth. Its se- 

 cond side is the prolongation of this tangent bent accordiug 

 to a vertical, and so on. According to this condition, the 

 author determines the equation of this curve, which is the 

 shortest that can be drawn between two points given on the 

 surface of the earth. 



The geodesic line is very proper for enabling us to be- 

 come acquainted with the true ligure of the earth. In fact, 

 we may conceive at every point of the surface of the earth 

 a tangent ellipsoid, and on which the geodesic measure- 

 ments, the longitudes, and latitudes, reckoninij from ih^ 

 points of contact, would be in a small extent the same as 

 at this surface. If the earth were an ellipsoid, it would 

 coincide with the tangent ellipsoid, which would be every 

 where the same ; but if this circumstance did not take place, 

 the tangent ellipsoid would vary from one country to an- 

 other} and these variations, which it is interesting to know, 

 could only be determined by geodesic admeasurements made 

 in different directions and in different countries. 



The surface of the earth being supposed to differ a little, 

 from the sphere, the author gives the equation of the geo- 

 desic line; and considering in the first place the case iu which 

 the tirst side of this line is parallel to the correopoiulinor. 

 plane of the celestial meridian, he deduces from it the 

 length of the arc comprised betweeii two given latitudes. 

 If the terrestrial spheroid be one of revolution, this arc and 

 the whole curve are iu one and the same plane, which isi: 

 that of the celesiial meridian. It varies from it when the 

 parallels are not circles ; so that the observation of this va- 

 riation may throw some light oi\ this^important point of the 

 figure of the earth. The author by a very dchcate atialvsis 

 shows, that if the first side of the geodesic line be parallel 

 to the corresponding plane of the celestial meridian, tiie dif- 

 ference of longitude of the two cxiremities of ilie measured 

 arc is equal t(j the azimuthal angle of the extreiiiitv of tlie 

 arc divided by the sine of the latitude. This very siu)ple 

 result is independent of the interior constitution of the 

 earth, and of the knowledge of its figure. It is of very 

 gieat importance iu this theory j since, if theazimuthal aufje 

 observed is buch that we cannot ascribe it to enors in the 

 observations, we njight conclude with certainty that the 

 earth is not a spheroid of revolution. Tlie author after- 

 ward? considers the case iu which the first side of the geo- 



dgsic 



