For the derivation, it is unnecessary to multiply by 100 to give percent as the 100 will cancel out later. This method seeks to determine m and c through minimizing the standard deviation of the fractional error. Again the actual division by (n-1) and taking of the square-root is superfluous:
Differentiate with respect to m for one equation then with respect to c for the second. Set the two derivatives each equal to zero to minimize the sum:
Solving the two simultaneous equations gives the following equations for m (the slope) and c (the constant):
I wrote a program in APPLESOFT BASIC to calculate the m and c pairs using an Apple IIc computer - - it was 1986! Dulong & Petit couldn't have done this in 1819 because, with Petit very sick with tuberculosis and dying the next year, they wouldn't have had time to do the tedious calculations without a modern computer. Finally, Mendeleyev's Periodic Table was not available until 1869 - - and, in 1869, this itself was a hard sell.
There is no correlation coefficient for this method yet. To tell just how well things are going, one can calculate the standard deviation of the experimental Y values with those calculated from the equation. However, this may come out, it will be known that no other linear equation will give a lower average error.