Let _k_ be the coefficient of friction. It is impossible to find any
value for _k_ which will satisfy the equation,
_F_ - _k R_ = 0,
for all the observed pairs of values of _F_ and _R_. We have then to
find the value for _k_ which, upon the whole, best represents the
experiments. _F_ - _k R_ is to be as near zero as possible for each
pair of values of _F_ and _R_.
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;
Σ _R₁ F₁_
whence _k_ = ————————————————.
Σ _R₁_²
The calculation of _k_ becomes simplified when (as is generally the
case in the tables) the loads _R₁_, _R₂_, &c., _Rₘ_ are of the form,
_N_, 2_N_, 3_N_ &c., _m N_.
In this case,
Σ _R₁_ _F₁_ = _N_ (_F₁_ + 2_F₂_ + 3_F₃_ + &c. + _mFₘ_).