_Let_ Q, R, S, T, &c. _represent any variable quantities, expressed in
terms of_ x _and_ y (_with given coefficients_), _and let_ q, r, s, t,
&c. _denote as many other quantities, expressed in terms of_ ẋ _and_ ẏ;
_It is proposed to find an equation for the relation of_ x _and_ y, _so
that the fluent of_ Qq + Rr + Ss + Tt, &c. _corresponding to a given
value of_ x (_or_ y), _may be a_ maximum _or_ minimum.
[Illustration]
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
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