text: Suppose that we have a sample consisting of 14 sheets described above, each one of which has an absorbance of 0.0222.  If we are able to estimate the absorbing power (the absorbance of a sample of the same thickness, but having no scatter) from the sample without knowing how many sheets are in the sample (as would be the general case), it would have the desirable property of being proportional to the thickness. In this case, we know that the absorbing power (scatter corrected absorbance) should be:  {14 x the absorbance of a single sheet}    .  This is the value we should have for the sample if the absorbance is to follow the law of Bouguer (often referred to as Beer's law).
  In the Table below, we see that the sample has the A,R,T values for the case of 14 sheets in the Table above.  Because of the presence of scatter, the measured absorbance of the sample would be: .  Then we calculate this for the half sample thickness using another of Benford's equations.   If ,  and  are known for a layer with thickness , the ART fractions for a layer with thickness of  are:
  In the line for half sample [S/2], we see the values which are the same as those for 7 layers in the Table above, as we expect.  Note that .  We desire to have the absorbance be linear with sample thickness, but we find when we multiply this value by 2, we get   , which is a significant departure from the previous estimate for the absorbing power.
  The next iteration of the formula produces the estimate for A,R,T for a quarter sample:  .  Note that this time the calculation corresponds to three and a half layers, a thickness of sample that cannot exist physically.
  Continuing for the sequentially higher powers of two, we see a monotonically increasing estimate.  Eventually the numbers will start jumping with round off error, but one can stop when getting a constant value to a specified number of significant figures.  In this case, we become constant to 4 significant figures at 0.3105, which is our estimate for the absorbing power of the sample.  This corresponds to our target value of 0.312 determined above.