2Q´q + q´Q + 2R´r + r´R, &c. - 2Q´´q + q´´Q - 2R´´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.
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.