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5. To illustrate these definitions: Suppose a point m be conceived to move from the position A, and to generate a line ar, by a motion any how regulated; and suppose the celerity of the point m, at any position P, to be such, as would, if from thence it should become or continue uniform, be sufficient to cause the point to describe, or pass uniformly over, the distance rp, in the given time allowed for the fluxion: then will the said line rp represent the fluxion of the fluent, or flowing line, AP, at that position.
6. Again, suppose the right line mn to move, from the position AB, continually parallel to itself,with any continued motion, so as to generate the fluent or flowing rectangle ABQP, while the point m describes the line AP: also, let the distance rp be taken, as before, to express the fluxion of the line or base AP; and complete the rectangle reqp. Then, like as pp is the fluxion of the line AP, so IS pq, the fluxion of the flowing parallelogram AQ: both these fluxions, or increments, being uniformly described in the same time.
7. In like manner, if the solid AERP be conceived to be generated by the plane PQR, moving from the position ABE, always parallel to itself, along the line AD ; and if pp denote the fluxion of the line AP: Then, like as the rectangle raqp, or PQ X PP, denotes the fluxion of the flowing rectangle ABqr, so also shall the fluxion of the variable solid, or prism Aberer, be denoted by the prism PQkrop, or the plane PR X PP. And, in both the last two cases, it appears that the fluxion of the generated rectangle, or prism, is equal to the product of the generating line, or plane drawn into the fluxion of the line along which it moves.
8. Hitherto the generating line, or plane, has been considered as of a constant and invariable magnitude; in which case the fluent, or quantity generated, is a rectangle, or a prism, the former being described by the motion of a line, and the latter by the motion of a plane. So, in like manner are other figures, whether plane or solid, conceived to be de
scribed by the motion of a Variable Magnitude, whether it
be carried by a motion perpendicular to the former and to describe the line re: let pq be another position of ra, indefinitely near to the former; and draw or parallel to AP. Now in this case there are several fluents, or flowing quantities, with their respective fluxions: namely, the line or fluent AP, the fluxion of which is rp or ar; the line or fluent ro, the fluxion of which is rq; the curve or oblique line aq, described by the oblique motion of the point a, the fluxion of which is aq; and lastly, the surface APQ, described by the variable line PQ, the fluxion of which is the rectangle Perp, or rQ X rp. In the same manner may any solid be conceived to be described, by the motion of a variable plane parallel to itself, substituting the variable plane for the variable line; in which case the fluxion of the solid, at any position, is represented by the variable plane, at that position, drawn into the fluxion. of the line along which it is carried.
9. Hence then it follows in general, that the fluxion of any figure, whether plane or solid, at any position, is equal to the section of it, at that position, drawn into the fluxion of the axis, or line along which the variable section is supposed to be perpendicularly carried: that is, the fluxion of the figure AQP, is equal to the plane PQXPp, when that figure is a solid, or to the ordinate PQ XPP, when the figure is a surface.
10. It also follows from the same premises, that in any curve or obliquë line aq, whose absciss is AP, and ordinate is PQ, the fluxions of these three form a small right-angled plane triangle agr; for or = pp is the fluxion of the absciss AP, qr the fluxion of the ordinate ro, and aq the fluxion of the curve or right line AQ. And consequently that, in any curve, the square of the fluxion of the curve, is equal to the
sum of the squares of the fluxions of the absciss and ordinate, when these two are at right angles to each other.
11. From the premises it also appears, that contemporaneous fluents, or quantities that flow or increase together, which are always in a constant ratio to each other, have their fluxions also in the same constant ratio, at every position. For, let AP and BQ be two contemporaneous fluents, described in the same time by the motion of the points P and Q, the contemporaneous positions being P, Q, and p, q; and let Ar be to BQ, B or Ap to вq, constantly in the ratio of 1 to n. Then
is n X AP=BQ, and n X Ap=Bq;
therefore, by subtraction, nXPp Qq;
that is, the fluxion pp: fluxion eq:: 1:n,
the same as the fluent AP: fluent BQ :: 1:n,
or, the fluxions and fluents are in the same constant ratio.
12. To apply the foregoing principles to the determination of the fluxions of algebraic quantities, by means of which those of all other kinds are assigned, it will be necessary first to premise the notation commonly used in this science, with some observations. As, first, that the final letters of the alphabet z, y, x, u, &c. are used to denote variable or flowing quantities; and the initial letters, a, b, c, d, &c. to denote constant or invariable ones: Thus, the variable base AP of the flowing rectangular figure ABQP, in art. 6, may be represented by x; and the invariable altitude PQ, by a: also, the variable base or absciss AP, of the figures in art. 8, may be represented by x, the variable ordinate PQ, by y; and the variable curve or line aq, by z.
Secondly, that the fluxion of a quantity denoted by a single letter, is represented by the same letter with a point over it: Thus, the fluxion of x is expressed by x, the fluxion of y by by, y, and the fluxion of z by ż. As to the fluxions of constant or invariable quantities, as of a, b, c, &c. they are equal to nothing, because they do not flow or change their magnitude.
Thirdly, that the increments of variable or flowing quantities, are also denoted by the same letters with a small over them: Thus, the increments of x, y, z, are x, y, z'.
13. From these notations, and the foregoing principles, the quantities and their fluxions, there considered, will be denoted as below. Thus, in all the foregoing figures, put
the variable or flowing line
in art. 8, the variable ordinate
AP = X,
PQ = α,
P Q = y,
Then shall the several fluxions be thus represented, namely,
x= Pp the fluxion of the line AP,
ax = Paqp the fluxion of abqr in art. 6,
@q=√(x2 + y2) the fluxion of aq; and
=Pr the fluxion of the solid in art. 7, if a denote the
nx = BQ in the figure to art. 11, and
eg the fluxion of the same.
14. The principles and notation being now laid down, we may proceed to the practice and rules of this doctrine; which consists of two principal parts, called the Direct and Inverse Method of Fluxions; namely, the direct method, which consists in finding the fluxion of any proposed fluent or flowing quantity; and the inverse method, which consists in finding the fluent of any proposed fluxion. As to the former of these two problems, it can always be determined, and that in finite algebraic terms; but the latter, or finding of fluents, can only be effected in some certain cases, except by means of infinite series.-First then, of
THE DIRECT METHOD OF FLUXIONS:
To find the Fluxion of the Product or Rectangle of two Varia-
15. Let ARQP, xy, be the flow-
pass through the intersection a of those lines, or the opposite
Now, the rectangle consists of the two trilinear spaces Arq, ARQ, of which, the
fluxion of the former is PQ XPp, or yx,
that of the latter is
RQ XRr, or xy, by art. 8;
therefore the sum of the two xy + xy, is the fluxion of the whole rectangle xy or arqp.
The Same Otherwise.
16. Let the sides of the rectangle x and y, by flowing, become x+x and y+y: then the product of these two, or xy +xy'+yx+x'y' will be the new or contemporaneous value of the flowing rectangle PR or xy: subtract the one value from the other, and the remainder, xy' + yx' + x'y', will be the increment generated in the same time as x'or y'; of which the last term x'y' is nothing or indefinitely small, in respect of the other two terms, because x and y are indefinitely small in respect of x and y; which term being therefore omitted, there remains xyyx for the value of the increment; and hence, by substituting and y, for x and y', to which they are proportional, there arises xy+yx for the true value of the fluxion of xy; the same as before.
17. Hence may be easily derived the fluxion of the powers and products of any number of flowing or variable quantities whatever; as of xyz, or uxyz, or vuxyz, &c. And first, for the fluxion of xyz: put p=xy, and the whole given fluent xyz=q, or q=xyz=pz. Then, taking the fluxions of q=pz, by the last article, they are qjz + pz; but · p2 + pz; but p➡xy, and so p=xy+xy by the same article: substituting therefore these values of p and instead of them, in the value of q this becomes q=xyz+xyz+xyz, the fluxion of xyz required; which is therefore equal to the sum of the products, arising from the fluxion of each letter, or quantity, multiplied by the product of the other two.
Again to determine the fluxion of uxyz, the continual product of four variable quantities; put this product, namely uxyz, or qu=r, where q=xyz as above. Then, taking the fluxions by the last article, qu+ qu; which, by substituting for q and their values as above, becomes
r=ux y z + w x y z +uxyz+uxyz, the fluxion of uxyz as required: consisting of the fluxion of each quantity, drawn into the products of the other three.