382 Astronomical and Nautical Collections. 



ance is fallacious ; for if there is any want of regularity in the 

 assumed grounds of calculation, some condition of the problem 

 must remain unfulfilled, if we imagine that we have represented 

 the whole of the observations. Thus in Lambert's method, 

 §. 65, we shall have determined the intervals of time correctly, 

 but the points found for the places of the comet will not lie in a 

 plane passing through the sun. According to Laplace, §. 67, 

 the time and distance of the perihelium, and the true anomalies 

 derived from them, will agree with the observations ; but the 

 heliocentric places, derived from them, will again deviate from a 

 great circle. The Newtonian determination of the longitude 

 of the node, and the inclination of the orbit, giving ?•', r", r'", 

 and k', k", correctly, and thence the time agreeing with the ob- 

 served interval, will indicate places not found in one and the 

 same parabola. In all these cases, therefore, we shall find not 

 one parabola, but strictly speaking three, differing more or less 

 from each other according to the accuracy of the observations, 

 or to the extent of the actual deviation of the comet from a pa- 

 rabola. In Laplace's method, the time and distance of the 

 perihelium only are the same for all three ; in Newton's, the 

 plane of the orbit only is the same ; in Lambert's, all the five 

 elements belong to three different parabolas. We must adhere 

 in practice to that orbit which agrees perfectly with the first and 

 third observations ; passing through the extreme points which 

 have been determined. 



§. 79. 



The condition that all the points of the orbit must lie in a 

 plane, passing through the sun's centre, is in itself the most 

 essential part of the theory : and hence the Newtonian mode 

 of correction has the advantage, since it satisfies this principal 

 condition : it has also the superior convenience of being imme- 

 diately applicable to the determination of the elements of an 

 elliptic orbit, if it should happen that a parabola is not sufficient 

 for the observations. 



§. 80. 



In order to determine if this is the case, we must compute 



