SECT. 4.] ON THE DETERMINATION OF ORBITS. 275 



often repeated. Accordingly, the first theory of perturbations will be constructed 

 upon those purely elliptical elements which have been approximately adjusted to 

 the observations ; a new orbit will afterwards be investigated, which, with the 

 addition of these perturbations, may satisfy, as far as practicable, the observa 

 tions. If this orbit differs considerably from the former, a second determination 

 of the perturbations will be based upon it, and the corrections will be repeated 

 alternately, until observations, elements, and perturbations agree as nearly as 

 possible. 



191. 



Since the development of the theory of perturbations from given elements is 

 foreign to our purpose, we will only point out here how an approximate orbit 

 can be so corrected, that, joined with given perturbations, it may satisfy, in 

 the best manner, the observations. This is accomplished in the most simple 

 way by a method analogous to those which we have explained in articles 124, 

 165, 188. The numerical values of the perturbations will be computed from the 

 equations,for the longitudes in orbit, for the radii vectores, and also for the helio 

 centric latitudes, for the times of all the observations which it is proposed to use, 

 and which can either be three, or four, or more, according to circumstances : for 

 this calculation the materials will be taken from the approximate elliptic ele 

 ments upon which the theory of perturbations has been constructed. Then two 

 will be selected from all the observations, for which the distances from the earth 

 will be computed from the same approximate elements : these will constitute the 

 first hypothesis, the second and third will be formed by changing these distances 

 a little. After this, in each of the hypotheses, the heliocentric places and the 

 distances from the sun will be determined from two geocentric places; from those, 

 after the latitudes have been freed from the perturbations, will be deduced the 

 longitude of the ascending node, the inclination of the orbit, and the longi 

 tudes in orbit. The method of article 110 with some modification is useful in 

 this calculation, if it is thought worth while to take account of the secular varia 

 tion of the longitude of the node and of the inclination. If p, ft , denote the 

 heliocentric latitudes freed from the periodical perturbations; \, If, the heliocen- 



