Analysis of the MBcanique Celeste of M* La Place. 475 

 orbits very much elongated, the author proceeds to determine 

 the elements of these orbits : he shows in the first, place how 

 we might deduce them from the circumstances of primitive 

 motion, if these circumstances were known ; and it is re- 

 markable that the direction of this motion does ne>t influence 

 the nature of the conic section. These researches produce 

 the discovery of the relation which exists between the trans- 

 verse axis of the orbit, the chord of the elliptic arc, the 

 sum .of the extreme radii vectores, and the time taken to de- 

 scribe this arc. 



As the observations do not make known the circumstances 

 attending the primitive motion of the celestial bodies, we 

 cannot determine from this supposition the elements of theif 

 orbits: it is necessary for this purpose to compare their re- 

 spective positions, observed at different epochs, with each 

 other : this is what we may do at all times with respect to 

 the planets, which we mav observe without interruption ; 

 but it is not the case with comets, which are only visible to 

 us in that part of their orbit which is nearest to the sun : it 

 is important, therefore, to be able to determine the elements 

 of the orbit of a comet from the circumstances attending its 

 appearance. In order to attain this, the author in the first 

 place gives converging formulae, which make known for a 

 given time, and according to any number of adjacent ob- 

 servations, the geocentric longitude and latitude of the co- 

 met, as well as their first and second differences divided by 

 the corresponding powers of the element of the time: he 

 shows that, by supposing these quantities to be known for a 

 given time in a system of bodies subjected to their mutual 

 attraction, we may easily, and without the assistance of in- 

 tegration, deduce therefrom the elements of the orbits. 



After having detailed these methods to the utmost extent 

 that is necessary, and given them all the perfection of which 

 they are susceptible ; attending also in a very simple manner 

 to the eccentricity of the terrestrial orbit, the author applies 

 them to the case of Nature, in which the orbits of comets 

 are ellipses greatly elongated, which are sensibly confounded 

 with the parabola towards the perihelion, which admits of 

 our considering their transverse axes as infinite : this cir^ 



