Let _A E_, _A F_, and _A G_, denote any three values of the quantity _x_, having indefinitely small _equi-differences E F_, _F G_; and let _E L_, _F M_, and _G N_, (perpendicular to _A G_) be the respective values of _y_, corresponding thereto; and, supposing _EF_ (= _FG_ = _ẋ_) to be denoted by _e_, let _c M_ and _d N_ (the successive values of _ẏ_) be represented by _u_ and _w_. Moreover, supposing _P´p´_ and _P´´p´´_ to be ordinates at the middle points _P´ P´´_, between _E_, _F_ and _F_, _G_, let the former (_P´p´_) be denoted α, and the latter (_P´´p´´_) by β; putting _A P´_ = _a_ and _A P´´_ = _b_. Then, if _a_ and α (the mean values of _x_ and _y_, between the ordinates _E L_ and _F M_) be supposed to be substituted for _x_ and _y_, in the given quantity _Qq_ + _Rr_ + _Ss_ + _Tt_, _&c._ and if, instead of _ẋ_ and _ẏ_, their equals _e_ and _u_ be also substituted, and the said (given) quantity, after such substitution, be denoted by _Q´q´_ + _R´r´_ + _S´s´_ + _T´t´_, _&c._ it is then evident, that this quantity _Q´q´_ + _R´r´_ + _S´s´_ + _T´t´_, _&c._ will express so much of the whole required fluent, as is comprehended between the ordinates _E L_ and _F M_, or as answers to an increase of _E F_ in the value of _x_. And thus, if _b_ and β be conceived to be wrote for _x_ and _y_, _e_ for _ẋ_, and _w_ for _ẏ_, and the quantity resulting be denoted by _Q´´q´´_ + _R´´r´´_ + _S´´s´´_ + _T´´t´´_, _&c._ this quantity will, in like manner, express the part of the required fluent corresponding to the interval _F G_. Whence that part answering to the interval _E G_ will consequently be equal to _Q´q´_ + _R´r´ &c._ + _Q´´q´´_ + _R´´r´´ &c._ But it is manifest, that the whole required fluent cannot be a _maximum_ or _minimum_, unless this part, supposing the bounding ordinates _E L_, _G N_ to remain the same, is also a _maximum_ or _minimum_. Hence, in order to determine the fluxion of this expression (_Q´q´_ + _R´r´ &c. Q´´q´´_ + _R´´r´´ &c._) which must, of consequence, be equal to nothing, let the fluxions of _Q´_ and _q´_ (taking α and _u_ as variable) be denoted by _̅Q_̇α and _̅qu⋅_; also let _̅R_̇α and _̅ru⋅_ denote the respective fluxions of _R´_ and _r´_; and let, in like manner, the fluxions of _Q´´, q´´, R´´, r´´, &c._ be represented by _̿Q_̇β, _̿q͘w_, _̿R_͘͘β͘ _̿rẇ_, _&c._ respectively. Then, by the common rule for finding the fluxion of a rectangle, the fluxion of our whole expression (_Q´q´_ + _R´r´ &c._ + _Q´´q´´_ + _R´´r´´ &c._) will be given equal to _Q´ ̅qu⋅_ + _q´ ̅Q_̇α + _R´ ̅ru⋅_ + _r´ ̅R_̇α _&c._ + _Q´´̿qẇ_ + _q´´ ̿Q_̇͘͘͘β + _R´´ ̿r͘w_ + _r´´ ̿R_̇β _&c._ = 0. But _u_ + _w_ being = _GN_ - _EL_, and β - α = (_GN_ - _EL_) ⁄ 2 (a constant quantity), we therefore have _ẇ_ = -_u͘_, and ̇β = ̇α: also _u_ being (= 2_rp´_) = 2α - 2_EL_, thence will _u͘_ = 2̇α: which values being substituted above, our equation, after the whole is divided by ̇α, will become