It is evident, in the first place, that the fluent of _Aẋ_ + _bBẋ_ + _cCẋ_ + _dDẋ, &c._ (_b, c, d, &c._ being any constant quantities whatever) must be a _maximum_, or _minimum_, in the proposed circumstance: and, if the relation of _x_ and _y_ be determined (_by the rule_), so as to answer this single condition (under all possible values of _b, c, d, &c._) it will also appear evident, that such relation will likewise answer and include all the other conditions propounded. For, there being in the general expression, thus derived, as many unknown quantities _b, c, d, &c._ (to be determined) as there are equations, by making the fluents of _Bẋ, Cẋ, Dẋ, &c._ equal to the values given; those quantities may be so assigned, or conceived to be such, as to answer all the conditions of the said equations. And then, to see clearly that the fluent of the first expression, _Aẋ_, cannot be greater than arises from hence (other things remaining the same) let there be supposed some other different relation of _x_ and _y_, whereby the conditions of all the other fluents of _Bẋ, Cẋ, Dẋ, &c._ can be fulfilled; and let, _if possible_, this new relation give a greater fluent of _Aẋ_ than the relation above assigned. Then, because the fluents _bBẋ, cCẋ, dDẋ, &c._ are given, and the same in both cases, it follows, according to this supposition, that this new relation must give a greater fluent of _Aẋ_ + _bBẋ_ + _cCẋ_ + _dDẋ, &c._ (under all possible values of _b, c, d, &c._) than the former relation gives: _which is impossible_; because (whatever values are assigned to _b, c, d, &c._) _that_ fluent will, it is demonstrated, be the greatest possible, when the relation of _x_ and _y_ is that above determined, by the General Rule. To exemplify, now, by a particular case, the method of operation above pointed out, let there be proposed the fluxionary quantity (_xⁿ yᵐ ẏᵖ_) ⁄ _ẋ⁽ᵖ ⁻ ¹⁾_; wherein the relation of _x_ and _y_ is so required, that the fluent, corresponding to given values of _x_ and _y_, shall be a _maximum_, or _minimum_. Here, by taking the fluxion, making _ẏ_ alone variable (_according to the rule_) and dividing by _ÿ_, we shall have (_pxⁿ yᵐ ẏ⁽ᵖ ⁻ ¹⁾_) ⁄ _ẋ⁽ᵖ ⁻ ¹⁾_ = υ. And, by taking the fluxion a second time, making _y_ alone variable, and dividing by _ẏ_, will be had (_mxⁿ y⁽ᵐ ⁻ ¹⁾ ẏᵖ_) ⁄ _ẋ⁽ᵖ ⁻ ¹⁾_ = ̇υ. Now from these equations to exterminate υ, let the latter be divided by the former; so shall _mẏ_ ⁄ _py_ = ̇υ ⁄ υ; and therefore _ay⁽ᵐ ⁄ ᵖ⁾_ = υ (_a_ being a constant quantity). From whence _y⁽ᵐ ⁄ ᵖ⁾ẏ_ = _(a ⁄ p)⁽¹ ⁄ ⁽ᵖ ⁻ ¹⁾⁾_ × _ẋx⁽⁻⁽ⁿ ⁄ ⁽ᵖ ⁻ ¹⁾⁾⁾_; and consequently (_p_ ⁄ (_m + p_)) × _y⁽⁽ᵐ + ᵖ⁾ ⁄ ᵖ⁾_ = _(a ⁄ p)⁽¹ ⁄ ⁽ᵖ ⁻ ¹⁾⁾_ × (_p_ - 1) ⁄ (_p_ - _n_ - 1) × _x_⁽⁽_ᵖ_ ⁻ _ⁿ_ ⁻ ¹⁾ ⁄ ⁽_ᵖ_ ⁻ ¹⁾⁾.