178 DETERMINATION OF AN ORBIT FROM [BOOK II. 



129. 



The five methods thus far explained lead, at once, to as many others which 

 differ from the former only in this, that the inclination of the orbit and the lon 

 gitude of the ascending node, instead of the distances from the earth, are taken 

 for x and y. The new methods are, then, as follows : 



I. From x and y, and the two extreme geocentric places, according to articles 

 74, 75, the heliocentric longitudes in orbit and the radii vectores are determined, 

 and, from these and the corresponding times, all the remaining elements ; from 

 these, finally, the geocentric place for the time of the middle observation, the 

 differences of which from the observed place in longitude and latitude will fur 

 nish X and Y. 



The four remaining methods agree in this, that all three of the heliocentric 

 longitudes in orbit and the corresponding radii vectores are computed from the 

 position of the plane of the orbit and the geocentric places. But afterwards: 



II. The remaining elements are determined from the two extreme places only 

 and the corresponding times ; with these elements the longitude in orbit and 

 radius vector are computed for the time of the middle observation, the differences 

 of which quantities from the values before found, that is, deduced from the geo 

 centric place, will produce X and Y: 



III. Or, the remaining dimensions of the orbit are derived from all three 

 heliocentric places (articles 82, 83,) into which calculation the times do not enter: 

 then the intervals of the times are deduced, which, in an orbit thus found, should 

 have elapsed between the first and second observation, and between this last 

 and the third, and their differences from the true intervals will furnish us with 

 X and Y: 



I V. The remaining elements are computed in two ways, that is, both by the 

 combination of the first place with the second, and by the combination of the 

 second with the third, the corresponding intervals of the times being used. These 

 two systems of elements being compared with each other, any two of the differ 

 ences may be taken for X and Y: 



V. Or lastly, the same double calculation is only continued to the values of 



