_F_ = _x_ + _y R_
shall be satisfied for all pairs of values of _F_ and _R_, but the best
values for _x_ and _y_ are those which make
(_F₁_ - _x_ - _y R₁_)² + (_F₂_ - _x_ - _y R₂_)² + &c.
+ (_Fₘ_ - _x_ - _y Rₘ_)²
a minimum.
Differentiating with respect to _x_ and _y_, and equating the
differential coefficients to zero, we have
Σ (_F₁_ - _x_ - _y R₁_) = 0,
Σ _R₁_(_F₁_ - _x_ - _y R₁_) = 0.
This gives two equations for the determination of _x_ and _y_.
Suppose, as is usually the case, the loads be of the form,
_N_, 2_N_, 3_N_, 4_N_ &c. _mN_,
and making
_A_ = _F₁_ + _F₂_ + _F₃_ + &c. + _Fₘ_
_B_ = _F₁_ +2_F₂_ + 3_F₃_ + &c. + _mFₘ_,
we have the equations
_m_(_m_ + 1)
_A_ - _mx_ - ————————————— _N y_ = 0,
2
_m_(_m_ + 1) _m_(_m_ + 1) (2_m_ + 1)
_B_ - ————————————— _x_ - ——————————————————————— _N y_ = 0.
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