FLUXION, in mathematics, denotes the velocity by which the fluents or flow ing quantities increase or decrease ; and may be considered as positive or nega tive, according as it relates to an incre ment or decrement.
The doctrine of fluxions, first invented by sir Isaac Newton, is of great use in the investigation of curves, and in the disco very of the quadratures of curvilinear spaces, and their ratifications. In this method, magnitudes are conceived to be generated by motion, and the velocity of the generating motion is the fluxion of the magnitude. Thus, the velocity of the point that describes a line is its fluxion, and measures its increase or decrease. When the motion of this point is uniform, its fluxion or velocity is constant, and may be measured by the space described in a given time ; but when the motion varies, the fluxion of velocity at any given point is measured by the space that would be described in a given time, if the motion was to be continued uniformly from that term.
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Thus, let the point in be conceived to A Ni 771 r --I--1 .......... .......... move from A, and generate the variable right line Am, by a motion any how regu lated; and let its velocity, when it arrives at any proposed position or point It, be such as would, was it to continue uniform from that point, be sufficient to describe the line Br, in the given time allotted for the fluxion, then will Ilp be the flux ion of the variable line A m, in the term or point It The flexion of a plain surface is con ceived in like manner, by supposing a given right line me (Plate V. Miscel. fig. 6) to move parallel to itself, in the plane of the parallel and moveable lines AF and BG : for if; as above, Itr be taken to ex• press the fluxion of the line A in, and the rectangle It r e S be completed; then that rectangle, being the space which would be uniformly described by the generating line M n, in the time that A in would be uniformly increased by in r, is therefore the fluxion of the generated rectangle 11 in, in that position.
if the length of the generating line Pi n continually varies, the fluxion of the area will still be expounded by a rectangle on -der that line, and the fluxion of the ab sciss or base : for let the curvilinear space A n ni (fig. 9,) be generated by the continual and parallel motion of the va riable line m n; and let It r be the fluxion of the base or absciss A na, as before, then the rectangle BraS will be the fluxion of the generated space A in n. Because, if the length and velocity of the gene rating line m n were to continue invaria ble from the position R. S, the rectangle 11 r e S would then he uniformly gene rated with the very velocity wherewith it begins to be generated, or with which the space A in n is increased in that posi tion.
FLuxions, notation of of invariable quantities, or those which neither increase -nor decrease, are represented by the first tters of the alphabet, • as a, b, c, d, &c, nd the variable or flowing quantities by e last letters, as zo, .r, y, z : thus, the iameter of a given circle may be de loted by a; and the sine of any arch thereof, considered as variable, by x. 1 Flie fluxion of a quantity, represented by r single letter, is expi essed by the same :letter with a dot or full point over it : thus, the fluxion of .x is represented by X, and that of y by y. And, • because these fluxions are themselves often va riable quantities, the velocities with which they either increase or decrease are the fluxions of the former fluxions, which may be called second fluxions, and are denoted by the same letters with two dots over them, and so on to the third, fourth, &c. fluxions. The whole doctrine of fluxions consists in solving the two following problems, viz. From the fluent, or variable flowing quantity given, to find the fluxion; which constitutes what is called the di rect method of fluxions. 2. From the fluxion given, to find the fluent, or flow ing quantity ; which makes the inverse method of fluxions.