Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/138

122 distance of the body from the foot of the perpendicular from the sun upon the line of sight. If we set  equations (8) become  Let us also set, for brevity,

Then $$\mathfrak{S}_{1}, \mathfrak{S}_{2}, \mathfrak{S}_{3}$$ may be regarded as functions respectively of $$\rho_{1}, \rho_{2}, \rho_{3},$$ therefore of $$q_{1}, q_{2}, q_{3},$$ and if we set and  we shall have  To determine the value of $$\mathfrak{S}',$$ we get by differentiation  But by (11)  Therefore

Now if any values of $$q_{1}, q_{2}, q_{3}$$ (either assumed or obtained by a previous approximation) give a certain residual $$\mathfrak{S}$$ (which would be zero if the values of $$q_{1}, q_{2}, q_{3}$$ satisfied the fundamental equation), and we wish to find the corrections $$\Delta q_{1}, \Delta q_{2}, \Delta q_{3}$$ which must be added to $$q_{1}, q_{2}, q_{3}$$ to reduce the residual to zero, we may apply equation (15) to these finite differences, and will have approximately, when these differences are not very large,