276 ON THE DETERMINATION OF ORBITS, [BOOK II. 



trie longitudes; Q, & -j- J, the longitudes of the ascending node; i,i-\-d, the 

 inclinations of the orbit ; the equations can be conveniently given in the follow 

 ing form : 



tan ft = tan i sin (A. & ), 



tun i ,/ . . / / j *~k \ 



,. i ,. tan a = tan t sin (A A Q), 



tan (i -\- 8) v 



This value of - r-^- acquires all the requisite accuracy by substituting an 

 approximate value for i: i and Q, can afterwards be deduced by the common 

 methods. 



Moreover, the sum of the perturbations will be subtracted from the longitudes 

 in orbit, and also from the two radii vectores, in order to produce purely elliptical 

 values. But here also the effect, which the secular variations of the place of the 

 perihelion and of the eccentricity exert upon the longitude in orbit and radius 

 vector, and which is to be determined by the differential formulas of Section I. 

 of the First Book, is to be combined directly with the periodical perturbations, 

 provided the observations are sufficiently distant from each other to make it 

 appear worth while to take account of it. The remaining elements will be deter 

 mined from these longitudes in orbit and corrected radii vectores together with 

 the corresponding times. Finally, from these elements will be computed the 

 geocentric places for all the other observations. These being compared with the 

 observed places, in the manner we have explained in article 188, that set of 

 distances will be deduced, from which will follow the elements satisfying in the 

 best possible manner all the remaining observations. 



192. 



The method explained in the preceding article has been principally adapted 

 to the determination of the first orbit, including the perturbations : but as soon 

 as the mean elliptic elements, and the equations of the perturbations have both 

 become very nearly known, the most accurate determination will be very con 

 veniently made with the aid of as many observations as possible by the method 

 of article 187, which will not require particular explanation in this place. Now 

 if the number of the best observations is sufficiently great, and a great interval 



