154 VECTOR METHOD IN THE DETERMINATION OF ORBITS. 



quantity sought is of the second order, and the relative error is of 

 the third order in the general case, or the fourth for equal intervals. 

 It is precisely this error which is most important in the case of 

 elliptic orbits. 



It will be observed that the accuracy of the expressions for the 

 ratios [ryrj : [r 2 r s ] : [r^] affords no measure of the accuracy of the 

 formula for the determination of elliptic orbits. 



I think that this hasty sketch will illustrate the convenience and 

 perspicuity of vector notations in this subject, quite independently of 

 any particular method which is chosen for the determination of the 

 orbit. What is the best method ? is hardly, I think, a question which 

 admits of a definite reply. It certainly depends upon the ratio of the 

 time intervals, their absolute value, and many other things. 



Yours very truly, 



J. WILLARD GIBBS. 



P.S. If we wish to use the curtate distances, with reference to the 

 ecliptic or the equator, let /o x be defined as the distance multiplied by 

 cosine (lat. or dec.), and ^i a s a vector of length secant (lat. or dec.). 

 For the most part the formulae will require no change, but the square 

 of 5i will b e sec 2 (lat. or dec.) instead of unity, so that the last terms 

 of (8) will have this factor. (&$ 2 $ 8 ) will then be Gauss' (0.1.2.), 

 whereas in my paper (3^i8k3k) * s L a g ran ge's (C'C"C'"}. 



J. W. G. 





