Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/167

Rh of the ecliptic, which wotdd make $$(\mathfrak{E}_{1} . \mathfrak{B}), (\mathfrak{E}_{2} . \mathfrak{B}), (\mathfrak{E}_{3} . \mathfrak{B})$$ vanish, except for exceedingly minute quantities depending on the latitude of the sun and the geocentric coordinates of the observatories, if these are included in $$\mathfrak{E}_{1}, \mathfrak{E}_{2}, \mathfrak{E}_{3}.$$

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 It might be made entirely symmetrical by writing $$-\tau_{2}$$ for $$\tau_{2}.$$ If an expression ending with $$t^3$$ had been used, we could still have satisfied two of the conditions relating to acceleration, and should have obtained  or  or