Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/169

It will be observed that all the formulaæ I, IIa, IIb, IIc, III may be expressed in the general form

${\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right){\mathfrak {R}}_{1}-\left(1+{\frac {B_{2}}{r_{2}^{3}}}\right){\mathfrak {R}}_{2}+\left(1+{\frac {B_{3}}{r_{3}^{2}}}\right){\mathfrak {R}}_{3}=0,$

except that the letters $B_{1},B_{2},B_{3}$ have different values in the different cases, some vanishing in the more simple formulæ. Moreover, if the values of $B_{1},B_{2},B_{3}$ have been calculated for I, the values for IIa, IIb, or IIc are found simply by subtraction of one of the numbers from the three. It is evident that IIb will hardly be useful except in special cases, as in the determination of a parabolic orbit in the failing case of Olbers' method, and then it would be a question whether it would not be better to determine the orbit from $\rho _{2}$ and $\rho _{3},$ or $\rho _{2}$ and $\rho _{1},$ using IIb or IIc.

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

$p_{1}{\mathfrak {R}}_{1}-p_{2}{\mathfrak {R}}_{2}+p_{3}{\mathfrak {R}}_{3}={\mathfrak {T}},$

which gives

$p_{1}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}-p_{2}{\mathfrak {R}}_{2}\times {\mathfrak {R}}_{3}={\mathfrak {T}}\times {\mathfrak {R}}_{3},$

${\frac {{\mathfrak {R}}_{3}\times {\mathfrak {R}}_{3}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{1}}{p_{2}}}-{\frac {{\mathfrak {T}}\times {\mathfrak {R}}_{3}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$

Now ${\frac {p_{1}}{p_{2}}}$ is my expression for the ratio of the triangles, and ${\frac {{\mathfrak {T}}\times {\mathfrak {R}}_{3}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}$ is its error. This is of the fourth order in general (since the denominator is of the first), and for equal intervals, of the fifth. The same is true of the two other ratios. Thus we have

${\frac {{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{2}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{3}}{p_{2}}}-{\frac {{\mathfrak {R}}_{1}\times {\mathfrak {T}}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$

Adding these equations and subtracting 1 [from both sides] we have

${\frac {{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{2}+{\mathfrak {R}}_{2}\times {\mathfrak {R}}_{3}-{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{1}+p_{3}-p_{2}}{p_{2}}}+{\frac {({\mathfrak {R}}_{3}-{\mathfrak {R}}_{1})\times {\mathfrak {T}}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$

Here the last term, which represents the error, is of the fifth order in general, or for equal intervals, of the sixth. But the