SECOND SECTION. 



DETERMINATION OF AN ORBIT PROM FOUR OBSERVATIONS, OF WHICH TWO 



ONLY ARE COMPLETE. 



164. 



WE have already, in the beginning of the second book (article 115), stated 

 that the use of the problem treated at length in the preceding section is lim 

 ited to those orbits of which the inclination is neither nothing, nor very small, 

 and that the determination of orbits slightly inclined must necessarily be based 

 on four observations. , But four complete observations, since they are equivalent 

 to eight equations, and the number of the unknown quantities amounts only to 

 six, would render the problem more than determinate : on which account it will 

 be necessary to set aside from two observations the latitudes (or declinations), 

 that the remaining data may be exactly satisfied. Thus a problem arises to 

 which this section will be devoted : but the solution we shall here give will ex 

 tend not only to orbits slightly inclined, but can be applied also with equal suc 

 cess to orbits, of any inclination however great. Here also, as in the problem of 

 the preceding section, it is necessary to separate the case, in which the approxi 

 mate dimensions of the orbit are already known, from the first determination 

 of a wholly unknown orbit : we will begin with the former. 



165. 



The simplest method of adjusting a known orbit to satisfy four observations 

 appears to be this. Let x, y, be the approximate distances of the heavenly body 

 from the earth in two complete observations : by means of these the correspond 

 ing heliocentric places may be computed, and hence the elements; after this, 

 (234) 



