SECT. 1.] THREE COMPLETE OBSERVATIONS. 171 



may be very small quantities, and it is evident that the differences between the 

 values of those functions corresponding to the system z=%, y = t], and those 

 which result from x , y = #, can be referred to a somewhat higher order 

 than the differences a, fj b , but the former values are X = , Y = t], and 

 the latter X a -\- A, Y = b -\- B, \vhence it follows, that a -\- A, b -\- B, are 

 much more exact values of x, y, than a, b. If the second hypothesis is based 

 upon these, the equations X= 0, Y= 0, are very frequently so exactly satisfied, 

 that it is not necessary to proceed any further ; but if not so, the third hypoth 

 esis will be formed in the same manner from the second, by making 



whence finally, if it is still not found sufficiently accurate, the fourth will be ob 

 tained according to the precept of article 120. 



123. 



We have supposed in what goes before, that the approximate values of the 

 unknown quantities x, y, are already had in some way. Where, indeed, the 

 approximate dimensions of the whole orbit are known (deduced perhaps from 

 other observations by means of previous calculations, and now to be corrected by 

 new ones), that condition can be satisfied without difficulty, whatever meaning we 

 may assign to the unknown quantities. On the other hand, it is by no means a 

 matter of indifference, in the determination of an orbit still wholly unkno\vn, 

 (which is by far the most difficult problem,) what unknown quantities we may 

 use ; but they should be judiciously selected in such a way, that the approximate 

 values may be derived from the nature of the problem itself. Which can be done 

 most satisfactorily, when the three observations applied to the investigation of 

 an orbit do not embrace too great a heliocentric motion of the heavenly body. 

 Observations of this kind, therefore, are always to be used for the first determina 

 tion, which may be corrected afterwards, at pleasure, by means of observations 

 more remote from each other. For it is readily perceived that the nearer the ob 

 servations employed are to each other, the more is the calculation affected by their 

 unavoidable errors. Hence it is inferred, that the observations for the first de- 



