_An Analytical Resolution of certain Equations of the Third, Fifth,
Seventh, Ninth Powers, and so on _ad Infinitum_, in finite Terms,
after the manner of _Cardan_'s Rules for Cubicks. By Mr. _A. Moivre_,
Transact. _Nᵒ 309_._
Let (_n_) be any Number, (_y_) an unknown Quantity, or Root of the
Equation, (_a_) a Quantity altogether known, or what they call
_Homogeneum Comparationis_: And let the Relation of these Quantities to
each other be exprest by the Equation.
_ny_ + ((_nn_ - 1)/(2 × 3))_ny_³ + ((_nn_ - 1)/(2 × 3)) ×
((_nn_ - 9)/(4 × 5))_ny_⁵ + ((_nn_ - 1)/(2 × 3)) ×
((_nn_ - 9)/(4 × 5)) × ((_nn_ - 25)/(6 × 7))_ny_⁷, _&c._ = _a_.
Its plain from the Nature of this Series, that if _n_ be any odd Number
(that is an Integer, it matters not whether Affirmative or Negative)
then the Series will Terminate, and the Equation arising will be one of
the above defin'd, whose Root is
(1) _y_ = ½[ⁿ√](√(1 + _aa_) + _a_) -
½/[ⁿ√](√(1 + _aa_) + _a_) or,
(2) _y_ = ½[ⁿ√](√(1 + _aa_) + _a_) -
½[ⁿ√](√(1 + _aa_) - _a_) or,
(3) _y_ = ½/[ⁿ√](√(1 + _aa_) - _a_) -
½[ⁿ√](√(1 + _aa_) - _a_) or,
(4) _y_ = ½/[ⁿ√](√(1 + _aa_) - _a_) -
½/[ⁿ√](√(1 + _aa_) + _a_)