The Auxillary Functions

  Note that the Correction Equations made use of two new vector data sets. We initially wanted to fit the functions φ₀=1, φ₁=exp(-λ₁t) and φ₂=exp(-λ₂t). The additional data sets correspond to the auxiliary functions φ₃=a₁t·exp(-λ₁t) and φ₄=a₂t·exp(-λ₂t) which determine the "directions" of the changes for changes in dλ₁ and dλ₂.

Computing the Best Fit for the Cooling Experiment

  If one knows an approximate fit for the the curve in the cooling experiment one can use Gauss' method of least squares to improve on the fit. The normal equations were given in his Theoria Motus.  The difficulty lies in finding a set of correction equatons. This can be accomplished by substituting a small correction to the set of parameters.

(tV) is a vector whose components are the individual products t_kV_k. One can then find the changes to the deviations.

One ends up with the following set of correction equatons.

One can use these equations to improve on an approximate fit by feedback of the corrected parameters.

With the same data as before,

One gets the following fit.

Supplemental (Jun 17): Gauss introduced the square bracket notation for the normal equations.