General Proposition.
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.
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 Rȧ 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, qẇ, R ̇β 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´´rẇ + r´´R ̇β &c. = 0.
But u + w being = GN - EL, and β - α = GN - EL ⁄ 2 (a constant quantity), we therefore have ẇ = -u⋅, and ̇β = ̇α: also u being (= 2rp´) = 2α - 2EL, thence will u⋅ = 2 ̇α: which values being substituted above, our equation, after the whole is divided by ̇α, will become
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