A More General LS Formula for Estimating a Common Factor

  One can generalize the formula in the last post to find the best estimate of a common factor.

Using this formula gives a better estimate for the wavenumber limit of the Balmer series as the following calculation shows. The formula gives 2744.8 while the average for the individual estimates is 2744.6 and the rms errors are 1.14 and 1.13 respectively.

The improvement over an average is marginal and doesn't appear to compensate for a scaling error.

Supplemental (Jul 8): Huggins adjusted his spectroscope for minimum deviation for the H line corresponding to 4340 Å or n=5. This looks like a better fit for the wavenumbers with an rms err of 0.46 which could result from rounding to the nearest Ångström.

This rescaling brings the estimate for the limit of the wavelengths to within a quarter of an Ångström of the currently accepted value.

Supplemental (Jul 8): The deflection of light by a prism depends on the index of refraction n and the wavelength λ. Shorter wavelengths are bent more than longer ones so blue light is bent more than red. The deviation from linearity also increases as the wavelength decreases so this might explain the need to rescale the spectroscope readings. The changes are in the right direction but I haven't compare the observed errors with calculated errors.