SECT. 1.] THREE COMPLETE OBSERVATIONS. 179 



the quantity denoted by t/, in article 91, and then the expressions given in the 

 preceding article for X and Y, are adopted. 



In order that the last four methods may be safely used, the places of the earth 

 for all three of the observations must not be very near the node of the orbit : on 

 the other hand, the use of the first method only requires, that this condition may 

 exist in the two extreme observations, or rather, (since the middle place may be 

 substituted for either of the extremes,) that, of the three places of the earth, 

 not more than one shall lie in the vicinity of the nodes. 



130. 



The ten methods explained from article 124 forwards, rest upon the assump 

 tion that approximate values of the distances of the heavenly body from the 

 earth, or of the position of the plane of the orbit, are already known. \\ hen 

 the problem is, to correct, by means of observations more remote from each other, 

 the dimensions of an orbit, the approximate values of which are already, by 

 some means, known, as, for instance, by a previous calculation based on other 

 observations, this assumption will evidently be liable to no difficulty. But it does 

 not as yet appear from this, how the first calculation is to be entered upon when 

 all the dimensions of the orbit are still wholly unknown : this case of our problem 

 is by far the most important and the most difficult, as may be imagined from 

 the analogous problem in the theory of comets, which, as is well known, has 

 perplexed geometers for a long time, and has given rise to many fruitless 

 attempts. In order that our problem may be considered as correctly solved, that 

 is, if the solution be given in accordance with what has been explained in the 

 119th and subsequent articles, it is evidently requisite to satisfy the following 

 conditions : First, the quantities x, y, are to be chosen in such a manner, that 

 we can find approximate values of them from the very nature of the problem, at 

 all events, as long as the heliocentric motion of the heavenly body between the 

 observations is not too great. Secondly, it is necessary that, for small changes in 

 the quantities x, y, there be not too great corresponding changes in the quantities 

 to be derived from them, lest the errors accidentally introduced in the assumed 

 values of the former, prevent the latter from being considered as approximate. 



