150 



Astronomical and Nautical Collections. 



§ 55. 

 Supposing us to be in possession of a long series of ob- 

 servations, we may proceed immediately to employ the most 

 distant of them for the correction of the elements, in a manner 

 which will hereafter be explained. But if, as often happens, 

 the comet has only been seen for two or three weeks, it will be 

 sufficient to correct the errors depending on the mode of com- 

 putation, by a process which is very easy and convenient : and 

 in this case it will be best to begin with distances somewhat 

 remote from each other, for instance, at an interval of 12, 14, 

 or 16 days, especially when the apparent distance of the comet 

 from the sun is not too small. 



§ 56. 

 It has often been observed that our method would he ma- 

 thematically correct if the revolving radii actually divided the 

 chords of the orbits in the precise proportion of the times ; for 

 then in fact we should have correctly f"'=:Mg' : since, however, 

 this is seldom exactly true, we have in reality §'"= (M + u) 

 l' + /t; and knoAving the elements pretty nearly, we may now 

 find the value of the corrections v and h. 



§57. 

 For the orbit of the earth, we have ad : t/c=R' sin (A" — A) 

 : R'" sin (A'"— A"); [since Sa=:R' and Sc=:R"', and the sines 

 of the angles at d are the same in the two triangles compared]. 



