Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/170

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 $[r_{1}r_{2}]:[r_{2}r_{3}]:[r_{1}r_{3}]$ 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 $\rho _{1}$ be defined as the distance multiplied by cosine (lat. or dec), and ${\mathfrak {F}}_{1}$ as a vector of length secant (lat. or dec). For the most part the formulae will require no change, but the square of ${\mathfrak {F}}_{1}$ will be $\sec ^{2}$(lat. or dec.) instead of unity, so that the last terms of (8) will have this factor. (${\mathfrak {F}}_{1}{\mathfrak {F}}_{2}{\mathfrak {F}}_{3}$) will then be Gauss' (0.1.2.), whereas in my paper (${\mathfrak {F}}_{1}{\mathfrak {F}}_{2}{\mathfrak {F}}_{3}$) is Lagrange's ($C'C''C'''$ ).

J. W. G.