In accordance with the principle of least squares, it is well known to mathematicians, the best value of k is that which makes
(F₁ - k R₁)² + (F₂ - k R₂)² + &c. + (Fₘ - k Rₘ)²
a minimum where f₁ and r₁, f₂ and r₂ &c. are the simultaneous values of f and r in the several experiments.
In fact, it is easy to see that, if this quantity be small, each of the essentially positive elements,
(F₁ - k R₁)², &c.
of which it is composed, must be small also, and that therefore
F - k R
must always be nearly zero.
Differentiating the sum of squares and equating the differential coefficient to zero, we have according to the usual notation,
Σ R₁ (F₁ - k R₁) = 0;