« PreviousContinue »
THE DOCTRINE OF FLUXIONS.
DEFINITIONS AND PRINCIPLES.
Art. 1. IN the Doctrine of Fluxions, magnitudes or quantities of all kinds are considered, not as made up of a number of small parts, but as generated by continued motion, by means of which they increase or decrease. As, a line by the motion of a point; a surface by the motion of a line; and a solid by the motion of a surface. So likewise, time may be considered as represented by a line, increasing uniformly by the motion of a point. And quantities of all kinds whatever, which are capable of increase and decrease, may in like manner be represented by geometrical magnitudes, conceived to be generated by motion.
2. Any quantity thus generated, and variable, is called a Fluent, or a Flowing Quantity. And the rate or proportion according to which any flowing quantity increases, at any position or instant, is the Fluxion of the said quantity, at that position or instant: and it is proportional to the magnitude by which the flowing quantity would be uniformly increased in a given time, with the generating celerity uniformly continued during that time.
3. The small quantities that are actually generated, produced, or described, in any small given time, and by any continued motion, either uniform or variable, are called Increments.
4. Hence, if the motion of increase be uniform, by which increments are generated, the increments will in that case be proportional, or equal, to the measures of the fluxions: but if the motion of increase be accelerated, the increment so generated, in a given finite time, will exceed the fluxion: and if it be a decreasing motion, the increment, so generated, will be less than the fluxion. But if the time be indefinitely small, so that the motion be considered as uniform for that instant; then these nascent increments will always be proportional, or equal, to the fluxions, and may be substituted instead of them, in any calculation.
A P p
5. To illustrate these definitions: Suppose a point m be conceived to move from the position A, and to generate a line AP, by a motion any how regulated; and suppose the celerity of the point w, 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 Pp, in the given time allowed for the fluxion: then will the said line pp represent the fluxion of the fluent, or flowing line, AP, at that position.
Q 1 C
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 ppbe taken, as before, to express the fluxion of the line or base AP; and complete the rectangle Poqp. Then, like as p 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 Poqp, or PQ × Pp, de
notes the fluxion of the flowing rectangle ABQP, so also shall the fluxion of the variable solid, or prism ABERQP, be denoted by the prism PQRrqp, or the plane PR × PP. And, in both these 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, con
ceived to be described by the motion of a Variable Magnitude, whether it be a line or a plane. Thus, let a variable line PQ be carried by a parallel motion along AP; or while a point P is carried along, and describes the line
4 4 4
AP, suppose another point a to be carried by a motion perpendicular to the former, and to describe the line PQ: let py be another position of pa, 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 pp or or; the line or fluent PQ, the fluxion of which is rq; the curve or oblique line AQ, described by the oblique motion of the point 2, the fluxion of which is oq; and lastly, the surface APQ, described by the variable line Pa, the fluxion of which is the rectangle rarp, or pa
Pp. 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 pq × PP, when that figure is a solid, or to the ordinate PQ × PP, when the figure is a surface.
10. It also follows from the same premises, that in any curve, or oblique line, AQ, whose absciss is AP, and ordinate is Po, the fluxions of these three form a small rightangled plane triangle aqr; for or Pp is the fluxion of the absciss AP, qr the fluxion of the ordinate PQ, and og 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 AP be to BQ, or Ap to Bq, constantly in the ratio of 1 to 22. Then is n X AP = BQ, and n x Ap = Bq;
therefore, by substraction, n x
pp = Qq;
that is, the fluxion - Pp fluxion oq :: 1 : n, the same as the fluent AP : fluent BQ :: 1 : n; or, the fluxions and fluents are in the same constant ratio. But if the ratio of the fluents be variable, so will that of the fluxions be also, though not in the same variable ratio with the former, at every position.
12. To apply the foregoing principles to the deternination 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 ; and the invariable altitude ro, by a: also, the variable base or absciss AP, of the figures in art. 8, may be represented by , 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 Ausion of y by ý, and the fluxion of z by 2. 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', ý, 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. 6, the constant line in art. 8, the variable ordinate also, the variable line or curve
PQ = a,
AQ = 2:
Then shall the several fluxions be thus represented, namely,
ax = Paqp the fluxion of ABQP in art. 6,
Qq = (√/ x*+j) the fluxion of AQ; and
Pr the fluxion of the solid in art. 7, if a denote the constant generating plane PQR; also,
nr=BQ in the figure to art. 11, and
nx = aq 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 Variable Quantities.
15. LET ARQP, = xy, be the flowing or variable rectangle, generated by two lines PQ and RQ, moving always perpendicular to each other, front the positions AR and AP; denoting the one by r, and the other by y; supposing x and y to be so related, that
the curve line ao may always pass through the intersection a of those lines, or the opposite angle of the rectangle.