Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/143

and therefore from $r_{1},r_{2},r_{3}$ and the given intervals. For our fundamental equation, which may be written

$n_{1}{\mathfrak {R}}_{1}-n_{2}{\mathfrak {R}}_{2}+n_{3}{\mathfrak {R}}_{3}=0,$

(33)

indicates that we may form a triangle in which the lengths of the sides shall be $n_{1}r_{1},n_{2}r_{2},$ and $n_{3}r_{3}$ (let us say for brevity, $s_{1},s_{2},s_{3}$), and the directions of the sides parallel with the three heliocentric directions of the body. The angles opposite $s_{1}$ and $s_{3}$ will be respectively $v_{3}-v_{2}$ and $v_{2}-v_{1}.$ We have, therefore, by a well-known formula,

$\tan {\frac {v_{3}-v_{2}}{2}}=$	${\sqrt {\frac {(s_{1}-s_{2}+s_{3})(s_{1}+s_{2}-s_{3})}{(s_{1}+s_{2}+s_{3})(-s_{1}+s_{2}+s_{3})}}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$. (34)
$\tan {\frac {v_{2}-v_{1}}{2}}=$	${\sqrt {\frac {(-s_{1}+s_{2}+s_{3})(s_{1}-s_{2}-s_{3})}{(s_{1}+s_{2}+s_{3})(s_{1}+s_{2}-s_{3})}}}$

As soon, therefore, as the solution of our fundamental equation has given a sufficient approximation to the values of $r_{1},r_{2},r_{3}$ (say five- or six-figure values, if our final result is to be as exact as seven-figure logarithms can make it), we calculate $n_{1},n_{2},n_{3}$ with seven-figure logarithms by equations (2), and the heliocentric angles by equations (34).

The semi-parameter corresponding to these values of the heliocentric distances and angles is given by the equation

$p={\frac {n_{1}r_{1}-n_{2}r_{2}+n_{3}r_{3}}{n_{1}-n_{2}+n_{3}}}\cdot$

(35)

The expression $n_{1}-n_{2}+n_{3},$ which occurs in the value of the semi-parameter, and the expression $n_{1}r_{1}-n_{2}r_{2}+n_{3}r_{3},$ or $s_{1}-s_{2}+s_{3},$ which occurs both in the value of the semi-parameter and in the formulæ for determining the heliocentric angles, represent small quantities of the second order (if we call the heliocentric angles small quantities of the first order), and cannot be very accurately determined from approximate numerical values of their separate terms. The first of these quantities may, however, be determined accurately by the formula

$n_{1}-n_{2}+n_{3}={\frac {A_{1}B_{1}}{r_{1}^{2}}}+{\frac {B_{2}}{r_{2}^{3}}}+{\frac {A_{3}B_{3}}{r_{3}^{3}}}\cdot$

(36)

With respect to the quantity $s_{1}-s_{2}+s_{3},$ a little consideration will show that if we are careful to use the same value wherever the expression occurs, both in the formulæ for the heliocentric angles and for the semi-parameter, the inaccuracy of the determination of this value from the cause mentioned will be of no consequence in the process of correcting the fundamental equation. For although the logarithm of $s_{1}-s_{2}+s_{3}$ as calculated by seven-figure logarithms from $r_{1},r_{2},r_{3}$ may be accurate only to four or five figures, we may regard it as absolutely correct if we make a very small change in the value of one