Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/149

Rh

(This equation will generally be most easily solved by repeated substitutions.)

$$\Delta q_{1}, \Delta q_{2}, \Delta q_{3}$$ are to be added as corrections to $$q_{1}, q_{2}, q_{3}.$$ With the new values thus obtained the computation by equations $$\text{III}_{1}, \text{III}_{2}, \text{III}_{3}$$ are to be recommenced. Two courses are now open:

(a) The work may be carried on exactly as before to the determination of new corrections for $$q_{1}, q_{2}, q_{3}.$$

(b) The computations by equations $$\text{III}', \text{III}, \text{III}',$$ and $$\text{IV}$$ may be omitted, and the old values of $$a_{1}, b_{1}, c_{1}, a_{2},$$ etc., $$G,$$ and $$L$$ may be used with the new residuals $$\alpha, \beta, \gamma$$ to get new corrections for $$q_{1}, q_{2}, q_{3}$$ by the equations  where $$Dq_{2}$$ denotes the former correction of $$q_{2}.$$ (More generally, at any stage of the work, $$Dq_{2}$$ will represent the sum of all the corrections of $$q_{2}$$ which have been made since the last computation of $$a_{1}, b_{1},$$ etc.) So far as any general rule can be given, it is advised to recompute $$a_{1}, b_{1},$$ etc., and $$G$$ once, perhaps after the second corrections of $$q_{1}, q_{2}, q_{3},$$ unless the assumed values represent a fair approximation. Whether $$L$$ is also to be recomputed, depends on its magnitude, and on that of the correction of $$q_{2},$$ which remains to be made. In the later stages of the work, when the corrections are small, the terms containing $$L$$ may be neglected altogether.

The corrections of $$q_{1}, q_{2}, q_{3}$$ should be repeated until the equations