Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/166

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},$

the expressions in brackets denoting numbers derived from the approximate values already found. This is similar to or identical with the method of Fabritius, except that he combines with it the principle of interpolation (for the first value in the third "hypothesis"). As I found the approximation by this method sometimes slow or failing, notably in the case of Swift's comet, 1880 V, I tried the method published in my paper. Indeed, it may be said that the method of my paper was constructed to meet the exigencies of the case of the comets 1880 V.

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.^[1] 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)

To eliminate $\rho _{3}$ and $r_{3},$ multiply by ${\mathfrak {E}}_{3}\times {\mathfrak {F}}_{3}$ which gives

$A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)[({\mathfrak {E}}_{1}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})+\rho _{1}({\mathfrak {F}}_{1}{\mathfrak {E}}_{2}{\mathfrak {F}}_{3})]-\left(1-{\frac {B_{2}}{r_{2}^{3}}}\right)[({\mathfrak {E}}_{1}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})+\rho _{2}({\mathfrak {F}}_{2}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})]=0.$

(b)

When we have found $\rho _{1},r_{1},\rho _{2},r_{2}$ it is not necessary to eliminate any of them, and to save labor in forming the equation for $\rho _{3},r_{3},$ I should be inclined to take the components in (7) in the direction of one of the coordinate axes, choosing that one which is most nearly directed towards the third observed position. However, I will write

$A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)[({\mathfrak {E}}_{1}.{\mathfrak {B}})+\rho _{1}({\mathfrak {F}}_{1}.{\mathfrak {B}})]-\left(1-{\frac {B_{2}}{r_{2}^{3}}}\right)[({\mathfrak {E}}_{2}.{\mathfrak {B}})+\rho _{2}({\mathfrak {F}}_{2}.{\mathfrak {B}})]+A_{3}\left(1+{\frac {B_{3}}{r_{3}^{3}}}\right)[({\mathfrak {E}}_{3}.{\mathfrak {B}})+\rho _{3}({\mathfrak {F}}_{3}.{\mathfrak {B}})]=0,$

(c)

where ${\mathfrak {B}}$ may represent an axis of coordinates, or $({\mathfrak {E}}_{1}\times {\mathfrak {F}}_{1})$ which would give Fabritius' equation. It might be directed towards the pole

1. [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.]