372 Aslronomicul and Nautical Cullectiuns. 



distances from ihe earth to be a, b, c, he takes for the true dis- 

 tances a + X, b + y, and c + *, expresses with these the dis- 

 tances of the comet from the sun, and ail the chords concerned, 

 and compares these, by means of his theorem, with the observed 

 intervals of time : and omitting all the higher powers of x, y, 

 and z, he, of course, obtains their values by linear equations. 

 But this computation is not a little troublesome and tedious » 

 and as I can assert from experience, incomparably more so than 

 might be inferred at first sight from the examples given by 

 Lambert. 



It is far more convenient to choose two of the approximate 

 elements and to compare them with three observations, in order 

 to see if they agree more or less accurately with them ; and then 

 to compute the effect of small alterations in the elements on each 

 observation. Hence the errors of these two elements will be- 

 come known, and they may be corrected accordingly ; and from 

 their corrected value, the other elements of the orbit may be 

 determined or corrected. 



i 67. 



Laplace chooses for this purpose the time and distance of 

 the perihelium. He then assumes three hypotheses, which, if T 

 be the time of the perihelium, and it the distance, obtained by 

 the approximation, may be thus represented, 1 ; 1, tt ; 2 ; 1 + r, 

 TT ; 3 ; 1, w + s ; and on each of these hypotheses he computes 

 for the time of three of the remotest observations the differences 

 of the true anomalies, and the distances of the comet from the 

 sun. From the three distances, and the observed geocentric 

 longitudes and latitudes, he finds again, by a calculation not 

 very difficult, the differences of the true anomalies. If these 

 differences agree for one of the hypotheses, the time and dis- 

 tance supposed in it must be correct ; if not, we may obtain 

 from these three comparisons the true time and distance, in a 

 manner which will be explained in speaking of the Newtonian 

 method. 1 do not enter more particularly into the method at 



