# Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter VI

VI.
ON THE USE OF THE VECTOR METHOD IN THE DETERMINATION OF ORBITS
Letter to Dr. Hugo Buchholz, Editor of Klinkerfues' Theoretische Astronomie.
New Haven, October, 1898.
Dr. Hugo Buchholz,
My dear Sir,—The opinion of Fabritius on the comparative convenience of different methods is entitled to far more weight than mine, for I am no astronomer, and have calculated very few orbits, none, indeed, except for the trial of my own formulæ. The object of my paper was to show to astronomers, who are rather conservative (and with right, for astronomy is the oldest of the exact sciences), the advantage in the use of vector notations, which I had learned in Physics from Maxwell. This object could be best obtained, not by showing, as I might have done, that much in the classic methods could be conveniently and perspicuously represented by vector notations, but rather by showing that these notations so simplify the subject, that it is easy to construct a method for the complete solution of the problem. That the method given is the best possible, I certainly do not claim, but only that it is much better than I could have found without the use of vector notations. Some of the more obvious crudities in my paper have been corrected in that of Beebe and Phillips. Doubtless many more remain, even if the general method be preserved.
My first efforts, however, to solve the fundamental approximative equation were along the same lines which Fabritius has followed:—to set $r_{1}=r_{2}$ and $r_{3}=r_{2}$ in equation second of (2) of Fabritius, which will give $\rho _{2}$ and $r_{2}$, then to get $r_{1}$ from the first of (3) of Fabritius, and then $r_{3}$ either from equation second of (3) or from some other ​which would serve the purpose, and then to find better values of $\rho _{2},r_{2}$ by setting in equation second of (2)
* $r_{1}=\left[{\frac {r_{1}}{r_{2}}}\right]r_{2},$$r_{3}=\left[{\frac {r_{3}}{r_{2}}}\right]r_{2},$
In ordinary cases I think that the method of Fabritius may very likely be better than that which I published. The equations are very simply and perspicuously represented in vector notations. I shall use the notations of my paper, writing ${\bar {E}},{\bar {F}},$ etc., for German letters. To eliminate $\rho _{1}$ and $\rho _{3}$ from equation (7) in my paper, multiply directly by ${\mathfrak {F}}_{1}\times {\mathfrak {F}}_{3}.$ This gives
| $A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)({\mathfrak {E}}_{1}{\mathfrak {F}}_{1}{\mathfrak {F}}_{3})-\left(1-{\frac {B_{2}}{r_{2}^{3}}}\right)[({\mathfrak {E}}_{2}{\mathfrak {F}}_{1}{\mathfrak {F}}_{3})+\rho _{2}({\mathfrak {F}}_{2}{\mathfrak {F}}_{1}{\mathfrak {F}}_{3})]+A_{3}\left(1+{\frac {B_{3}}{r_{3}^{3}}}\right)({\mathfrak {E}}_{3}{\mathfrak {F}}_{1}{\mathfrak {F}}_{3})=0.$ | (a) |
$\rho _{3}$
$r_{3},$
${\mathfrak {E}}_{3}\times {\mathfrak {F}}_{3}$
| $A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)[({\mathfrak {E}}_{1}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})+\rho _{1}({\mathfrak {F}}_{1}{\mathfrak {E}}_{2}{\mathfrak {F}}_{3})]-\left(1-{\frac {B_{2}}{r_{2}^{3}}}\right)[({\mathfrak {E}}_{1}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})+\rho _{2}({\mathfrak {F}}_{2}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})]=0.$ | (b) |
$\rho _{1},r_{1},\rho _{2},r_{2}$
$\rho _{3},r_{3},$
| $A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)[({\mathfrak {E}}_{1}.{\mathfrak {B}})+\rho _{1}({\mathfrak {F}}_{1}.{\mathfrak {B}})]-\left(1-{\frac {B_{2}}{r_{2}^{3}}}\right)[({\mathfrak {E}}_{2}.{\mathfrak {B}})+\rho _{2}({\mathfrak {F}}_{2}.{\mathfrak {B}})]+A_{3}\left(1+{\frac {B_{3}}{r_{3}^{3}}}\right)[({\mathfrak {E}}_{3}.{\mathfrak {B}})+\rho _{3}({\mathfrak {F}}_{3}.{\mathfrak {B}})]=0,$ | (c) |
${\mathfrak {B}}$
$({\mathfrak {E}}_{1}\times {\mathfrak {F}}_{1})$
$({\mathfrak {E}}_{1}.{\mathfrak {B}}),({\mathfrak {E}}_{2}.{\mathfrak {B}}),({\mathfrak {E}}_{3}.{\mathfrak {B}})$
${\mathfrak {E}}_{1},{\mathfrak {E}}_{2},{\mathfrak {E}}_{3}.$
The equations (a), (b), (c), which are together equivalent to (7), I would solve as follows, almost in the same way as Fabritius, but relying a little more on interpolation, and less on the convergence of which he speaks, which in special cases may more or less fail.
Setting $r_{1}=r_{2}$ and $r_{3}=r_{2}$ in (a), which thus modified I shall call (a'), and solving this (a') by "trial and error," using $\rho _{2}$ as the independent variable, as soon as I have a value of $\rho _{2}$ which I think will give a residual of (a') of the same order of smallness as the effect of changing ${\frac {1}{r_{1}^{3}}}$ and ${\frac {1}{r_{3}^{3}}}$ into ${\frac {1}{r_{2}^{3}}},$ I determine from this value by (b) and (c), $r_{1}$ and $r_{3},$ and then find the residual of (a), using the values of $r_{1},r_{2},r_{3}$ derived all from the same assumed $\rho _{2}.$ Now using the last value of ${\frac {\Delta {\text{(residual)}}}{\Delta \rho _{2}}}$ in my previous calculations on (a') which indeed applies only roughly to the (a), I would get a value $\rho _{2}$ which I would use for the second "hypothesis" in (a). This will give a second residual in (a), which will enable me to make a more satisfactory interpolation. As many more interpolations may be made as shall be found necessary.
Some such method, which should perhaps be called the method of Fabritius, would, I think, in most cases probably be the best for solution of equation (7).
Of course I am quite aware that the merit of my paper, if any, lies principally in the fundamental approximation (1). I will add a few words on this subject.
The equation may be written more symmetrically
| ${\begin{aligned}{\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {\tau _{2}^{2}+\tau _{3}^{2}-3\tau _{1}^{2}}{24r_{1}^{3}}}\right){\mathfrak {R}}_{1}&-\left(1+{\frac {\tau _{3}^{2}+\tau _{1}^{2}-3\tau _{2}^{2}}{24r_{2}^{3}}}\right){\mathfrak {R}}_{2}\\&{\frac {\tau _{3}}{\tau _{2}}}\left(1+{\frac {\tau _{1}^{2}+\tau _{2}^{2}-3\tau _{3}^{2}}{24r_{3}^{3}}}\right){\mathfrak {R}}_{3}=0.\end{aligned}}$ | I |
$-\tau _{2}$
$\tau _{2}.$
$t^{3}$
| ${\frac {\tau _{1}}{\tau _{2}}}{\mathfrak {R}}_{1}-\left(1+{\frac {\tau _{1}^{2}-\tau _{2}^{2}}{6r_{2}^{2}}}\right){\mathfrak {R}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}\left(1+{\frac {\tau _{1}^{2}-\tau _{3}^{2}}{6r_{3}^{2}}}\right){\mathfrak {R}}_{3}=0,$ | IIa |
| ${\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {\tau _{2}^{2}-\tau _{1}^{2}}{6r_{1}^{2}}}\right){\mathfrak {R}}_{1}-{\mathfrak {R}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}\left(1+{\frac {\tau _{2}^{2}-\tau _{3}^{2}}{6r_{3}^{2}}}\right){\mathfrak {R}}_{3}=0,$ | IIb |
| ${\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {\tau _{3}^{2}-\tau _{1}^{2}}{6r_{1}^{2}}}\right){\mathfrak {R}}_{1}-\left(1+{\frac {\tau _{3}^{2}-\tau _{2}^{2}}{6r_{2}^{2}}}\right){\mathfrak {R}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}{\mathfrak {R}}_{2}=0.$ | IIc |
$t^{2}$
| ${\frac {\tau _{1}}{\tau _{2}}}{\mathfrak {R}}_{1}-\left(1-{\frac {\tau _{1}\tau _{3}}{2r_{2}^{3}}}\right){\mathfrak {R}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}{\mathfrak {R}}_{3}=0.$ | III |
* ${\frac {\tau _{1}}{\tau _{2}}}{\mathfrak {R}}_{1}-{\frac {{\mathfrak {R}}_{2}}{1+{\frac {\tau _{1}\tau _{3}}{2r_{2}^{3}}}}}+{\frac {\tau _{3}}{\tau _{2}}}{\mathfrak {R}}_{3}=0,$
Writing an equation analogous to III for the earth and subtracting from (7), Mem. Nat. Acad., we have
* ${\frac {\tau _{1}}{\tau _{2}}}\rho _{2}{\mathfrak {F}}_{1}-\left(1-{\frac {\tau _{1}\tau _{3}}{2r_{2}^{3}}}\right)\rho _{2}{\mathfrak {F}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}\rho _{3}{\mathfrak {F}}_{3}={\frac {\tau _{1}\tau _{3}}{2}}\left({\frac {1}{{\mathfrak {E}}_{2}^{3}}}-{\frac {1}{r_{2}^{3}}}\right){\mathfrak {E}}_{2},$
${\mathfrak {F}}_{1}\times {\mathfrak {F}}_{3}$
${\mathfrak {F}}_{2}\times {\mathfrak {E}}_{2},$
It is evident that in general the error in I is of the fifth order, in IIa, IIb, IIc of the fourth, and in III of the third. But for equal intervals, the error in I is of the sixth order, and in III of the fourth. And when $\tau _{2}^{2}+\tau _{3}^{2}-3\tau _{1}^{2}=0,$ IIa becomes identical with I, and its error is of the fifth order.
The same is true of IIc in the corresponding case. It follows that when the intervals are nearly as 5:8 we should use IIa or IIb instead of I. This will evidently abbreviate the solution given above as only one of the quantities $r_{1},r_{3},$ is to be used.
The formulæ IIa, IIb, IIc may also be obtained by the following method, which will show their relative accuracy.
The interpolation formula
* ${\frac {\tau _{1}}{\tau _{2}}}{\frac {{\mathfrak {R}}_{1}}{r_{1}^{3}}}-{\frac {{\mathfrak {R}}_{2}}{r_{2}^{3}}}+{\frac {\tau _{3}}{\tau _{2}}}{\frac {{\mathfrak {R}}_{3}}{r_{3}^{3}}}=0$
${\frac {\tau _{2}^{2}+\tau _{3}^{2}-3\tau _{1}^{2}}{24}}$
* ${\frac {\tau _{3}^{2}+\tau _{1}^{2}-3\tau _{2}^{2}}{24}}$or${\frac {\tau _{1}^{2}+\tau _{2}^{2}-3\tau _{3}^{2}}{24}}$
​It will be observed that all the formulaæ I, IIa, IIb, IIc, III may be expressed in the general form
* ${\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right){\mathfrak {R}}_{1}-\left(1+{\frac {B_{2}}{r_{2}^{3}}}\right){\mathfrak {R}}_{2}+\left(1+{\frac {B_{3}}{r_{3}^{2}}}\right){\mathfrak {R}}_{3}=0,$
$B_{1},B_{2},B_{3}$
$B_{1},B_{2},B_{3}$
$\rho _{2}$
$\rho _{3},$
$\rho _{2}$
$\rho _{1},$
Equations IIa and IIc are very appropriate for the determination of an elliptic orbit when the observed motion is nearly in the ecliptic, by means of four observations with intervals nearly in the ratio 5 : 8 : 5.
It is evident that the solution of (7) given above may be varied, in ways too numerous to mention, by the use of the simpler forms IIa, IIc, or III for II in the earlier stages of the work. This only involves changing the values of $B_{1},B_{2},B_{3}$ in (a), (b) and (c).
It is not correct to say that in my expressions for the ratios of the triangles the error is of the fifth order in general, or for equal intervals, of the sixth. If we write $p_{1},p_{2},p_{3},$ for the coefficients of ${\mathfrak {R}}_{1},{\mathfrak {R}}_{2},{\mathfrak {R}}_{3}$ in I, and ${\mathfrak {T}}$ for the error of the equation, we have exactly
* $p_{1}{\mathfrak {R}}_{1}-p_{2}{\mathfrak {R}}_{2}+p_{3}{\mathfrak {R}}_{3}={\mathfrak {T}},$
* $p_{1}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}-p_{2}{\mathfrak {R}}_{2}\times {\mathfrak {R}}_{3}={\mathfrak {T}}\times {\mathfrak {R}}_{3},$
* ${\frac {{\mathfrak {R}}_{3}\times {\mathfrak {R}}_{3}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{1}}{p_{2}}}-{\frac {{\mathfrak {T}}\times {\mathfrak {R}}_{3}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$
${\frac {p_{1}}{p_{2}}}$
${\frac {{\mathfrak {T}}\times {\mathfrak {R}}_{3}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}$
* ${\frac {{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{2}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{3}}{p_{2}}}-{\frac {{\mathfrak {R}}_{1}\times {\mathfrak {T}}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$
* ${\frac {{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{2}+{\mathfrak {R}}_{2}\times {\mathfrak {R}}_{3}-{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{1}+p_{3}-p_{2}}{p_{2}}}+{\frac {({\mathfrak {R}}_{3}-{\mathfrak {R}}_{1})\times {\mathfrak {T}}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$
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.
* ↑ [In which the preceding memoir, and also that of Beehe and Phillips referred to below, are translated.]
* ↑ [See Fabritius, W. "Ueber eine leichte Methode der Bahnbestimmung mit Zugrundgelegung des Princips von Gibbs," also "Weitere Anwendongen dee Gibbs'chen Princips," Astronomische Nachrichten, No. 3061, 3065 (1891)].
* ↑ ["The Orbit of Swift's comet, 1880 V, Determined by Gibbs's Vector Method," W. Beebe and A. W. Phillips. Gould's Astronomical Journal, vol. ix, Dec. 1889.]
* ↑ [In the remainder of the letter as here printed German capitals have been substituted for the ${\bar {E}},{\bar {F}},{\bar {P}}$ etc. of the original, thus making the notation uniform with that of the paper referred to.]