2_Q´ ̅q_ + _q´ ̅Q_ + 2_R´ ̅r_ + _r´ ̅R, &c._ - 2_Q´´ ̿q_ + _q´´ ̿Q_ - 2_R´´ ̿r_ + _r´ ̿R, &c._ = 0; or, _Q´´ ̿q_ - _Q´ ̅q_ + _R´´ ̿r_ - _R´ ̅r &c._ = (_q´ ̅Q_ + _q´´ ̿Q_) ⁄ 2 + (_r´ ̅R_ + _r´´ ̿R_) ⁄ 2, _&c._ But _Q´´ ̿q_ - _Q´ ̅q_, the excess of _Q´´ ̿q_ above _Q´ ̅q_, is the increment or fluxion (answering to the increment, or fluxion, _ẋ_) arising by substituting _b_ for _a_, β for α, and _w_ for _u_. Moreover, with regard to the quantities on the other side of the equation, it is plain, seeing the difference of _q´ ̅Q_ and _q´´ ̿Q_ is indefinitely little in comparison of their sum, that _q´ ̅Q_ may be substituted in the room of (_q´ ̅Q_ + _q´´ ̿Q_) ⁄ 2, _&c._ which being done, our equation will stand thus: _Flux. Q´ ̅q_ + _R´ ̅r &c._ = _q´ ̅Q_ + _r´ ̅R &c._ But _q´ ̅Q_ + _r´ ̅R &c._ represents (by the preceding notation) the fluxion of _q´Q´_ + _r´R´ &c._ (or of _Qq_ + _Rr &c._) arising by substituting α for _y_, making α alone variable, and casting off ̇α. If, therefore, that fluxion be denoted by ̇υ, we shall have _flux. Q´ ̅q_ + _R´ ̅r &c._ = ̇υ, and consequently _Q´ ̅q_ + _R´ ̅r &c._ = υ. But _Q´ ̅q_ + _R´ ̅r &c._ (by the same notation) appears to be the fluxion of _Q´q´_ + _R´r´ &c._ (or of _Qq_ + _Rr &c._) arising by substituting _u_ for _ẏ_, making _u_ alone variable, and casting off _̇u_. Whence the following GENERAL RULE. _Take the fluxion of the given expression_ (_whose fluent is required to be a_ maximum _or_ minimum) _making_ ẏ _alone variable; and, having divided by_ ÿ, _let the quotient be denoted by_ υ: _Then take, again, the fluxion of the same expression, making_ y _alone variable, which divide by_ ẏ; _and then this last quotient will be_ = ̇υ. When _ẏ_ is not found in the quantity given, υ will then be = 0; and, consequently, the expression for ̇υ, equal to nothing also. But if _y_ be absent, then will ̇υ = 0, and consequently the value of υ = a constant quantity. It is also easy to comprehend, that, instead of _ẏ_ and _y, ẋ_ and _x_ may be made successively variable. Moreover, should the case to be resolved be confined to other restrictions, besides that of the _maximum_ or _minimum_, such as, having a certain number of other fluents, at the same time, equal to given quantities, still the same method of solution may be applied, and that with equal advantage, if from the particular expressions exhibiting all the several conditions, one general expression composed of them all, with unknown (but determinate) coefficients, be made use of. In order to render this matter quite clear, let _A, B, C, D, &c._ be supposed to represent any quantities expressed in terms of _x, y_, and their fluxions, and let it be required to determine the relation of _x_ and _y_, so that the fluent of _Aẋ_ shall be a _maximum_, or _minimum_, when the cotemporary fluents of _Bẋ, Cẋ, Dẋ, &c._ are, all of them, equal to given quantities.