-f.......... r 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 R, be such as would, was it to continue uniform from that point, be sufficient to describe the line Rr, in the given time allotted for the fluxion, then will Rr be the fluxion of the variable line A m, in the term or point R The fluxion of a plain surface is conceived in like manner, by supposing a given right line mn (Plate V. Miscel. fig. 8) to move parallel to itself, in the plane of the parallel and moveable lines AF and BG: for if, as above, Rr be taken to express the fluxion of the line A m, and the rectangle Rrs S be completed; then that rectangle, being the space which would be uniformly described by the generating line mn, in the time that A m would be uniformly increased by mr, is therefore the fluxion of the generated rectangle Bm, in that position. : If the length of the generating line mn continually varies, the fluxion of the area will still be expounded by a rectangle under that line, and the fluxion of the absciss or base for let the curvilinear space A n m (fig. 9,) be generated by the continual and parallel motion of the variable line m n; and let Rr be the fluxion of the base or absciss A m, as before, then the rectangle Rr & S will be the fluxion of the generated space Am n. Because, if the length and velocity of the generating line m n were to continue invariable from the position RS, the rectangle Rrs S would then be uniformly generated with the very velocity wherewith it begins to be generated, or with which the space A mn is increased in that position. FLUXIONS, notation of, of invariable quantities, or those which neither increase nor decrease, are represented by the first letters of the alphabet, as a, b, c, d, &c. and the variable or flowing quantities by the last letters, as v, w, x, y, z: thus, the diameter of a given circle may be denoted by a; and the sine of any arch thereof, considered as variable, by r. The fluxion of a quantity, represented by a single letter, is expressed by the same letter with a dot or full point over it: VOL. V. thus, the fluxion of is represented by a, and that of y by y. And, because these fluxions are themselves often variable 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 direct method of fluxions. 2. From the fluxion given, to find the fluent, or flowing quantity; which makes the inverse method of fluxions. FLUXIONS, direct method of, the doctrine of this part of fluxions is comprised in these rules. 1. To find the fluxion of any simple variable quantity, the rule is to place a dot over it thus, the fluxion of x is, and of y, y Again, the fluxion of the compound quantity x+y, is itỷ, also the Auxion of x-y, is i—ÿ 2 To find the fluxion of any given power of a variable quantity, multiply the Aluxion of the root by the exponent of the power, and the product by that power of the same root, whose exponent is less by unity than the given exponent This rule. is expressed more briefly, in algebraical i characters, by n x n x. the fluxion of Thus the fluxion of x3 is × 3 × ×1 3; and the fluxion of a is x × 5 Xx+= 514 x. In the same manner the fluxion of aty is 7 ÿ × ay 6; for the quantity a being constant, y is the true fluxion of the root a+y Again, the flux. ion of a will be × 2 zi× az): for here a being put we have 2 z ż; and therefore 3x2 for the fluxion of x 3 (or u+z2) is= z ż√ a2 + z2 3 3 To find the fluxion of the product of several variable quantities, multiply the fluxion of each by the product of the rest of the quantities; and the sum of the products, thus arising, will be the fluxion sought. Thus, the fluxion of x y is x y ya; that of x y z is x y z + ÿ x z + ż x y; and that of xyz is vx y z + ż v y z + Z 5. When the proposed quantity is affected by a coefficient, or constant multiplicator, the fluxion found as above must be multiplied by that coefficient or multiplicator: thus the fluxion of 5 x3, is 15 x2 x; for the fluxion of x3 is 3 x', which multiplied by 5, gives 15 xx. And, in the very same manner, the fluxion of a xn will be na xn-1 x. Hence it appears, that whether the root be a simple or a compound quantity, the fluxion of any power of it is found by the following general Rule: Multiply by the index, diminish the index by unity, and multiply by the fluxion of the root. Thus the Auxion of -8 8 27 the fluxion of 4 x5 the fluxion of 24x5 and 4 of this kind to enter upon the second, third, &c. fluxions, we shall therefore proceed to FLUXIONS, inverse method of, or the manner of determining the fluents of given fluxions. If what is already delivered, concerning the direct method, be duly considered, there will be no great difficulty in conceiving the reasons of the inverse method; though the difficulties that occur in this last part, upon another account, are indeed vastly great. It is an easy matter, or not impossible at most, to find the fluxion of any flowing quantity whatever; but, in the inverse method, the case is quite otherwise; for, as there is no method for deducing the fluent from the fluxion a priori, by a direct investigation, so it is impossible to lay down rules for any other forms of fluxions than those particular ones, that we know, from the direct method, belong to such and such kinds of flowing quantities; thus, for example, the fluent of 2 x x is known to be x'; because, by the direct method, the fluxion of x is found to be 2 x : but the fluent of y is unknown, since no expression has been discovered that produces y for its fluxion. Be this as it will, the following rules are those used by the best mathematicians, for finding the fluents of given fluxions. 1. To find the fluent of any simple fluxion, you need only write the letters without the dots over them: thus, the fluent of is x, and that of a + by, is ax + by. 2. To assign the fluent of any power of a variable quantity, multiplied by the fluxion of the root; first divide by the fluxion of the root, add unity to the exponent of the power, and divide by the exponent so increased: for dividing the fluxion n xn-1 by x, it becomes nxn−1; and adding 1 to the exponent (n − 1) we have nx; which divided by n, gives an, the true fluent of n xn-1 . Hence, by the same rule, the fluent of 3 x will be x6 x3; that of 2 x5 x ; that of yÿ 3 3 ż 5 Having explained the manner of determining the first fluxions of variable quantities, it is unnecessary in a work er a ƒ x-n= = = a + = 4 4 a xl-n 1-n n and that of am+2m x 2m-12 am+ zmn mxn+1 Hitherto x and y are both supposed ; that of a × 23+ equal to nothing at the same time which will not always be the case: thus, for instance, though the sine and tangent of an arch are both equal to nothing, when the arch itself is so; yet the secant is then equal to the radius. It will there. fore be proper to add some examples, wherein the value of y is equal to nothing, when that of x is equal to any given quan= x2x, tity a. Thus, let the equation y be proposed; whereof the fluent first In assigning the fluents of given fluxions, it ought to be considered whether the flowing quantity, found as above, requires the addition or subtraction of some Constant quantity, to render it complete thus, for instance, the fluent of n xn-1 may be either represented by xn or by x ±a; for a being a constant quantity, the fluxion of na, as well as of x", is nxn-1i. Hence it appears that the variable part of a fluent only can be assigned by the common method, the constant part being only assignable from the particular nature of the problem. Now to do this the best way is, to consider how much the variable part of the fluent, first found, differs from the truth, when the quantity which the whole fluent ought to express is equal to nothing; then that difference, added to, or subtracted from, the said variable part, as occasion requires, will give the To make this fluent truly corrected. plainer by an example or two, let y a+ 13 X x. Here we first find y 0, then "+x+ a + x 4 ; but when y 4 a 4. 3. To find the fluents of such fluxionary expressions as involve two or more variable quantities, substitute, instead of such fluxion, its respective flowing quantity; and, adding all the terms together, divide the sum by the number of terms, and the quotient will be the fluent. Thus, xy+xy y ï y+ỷ 2 sincer, by hypothesis, the fluent of y+x=x y + x y 2 x ! 2 = xy; and the fluent of x y z + yx z+ży x =x y 2. But it seldom happens that these kinds of fluxions, which involve two variable quantities in one term, and yet admit of known and perfect fluents, are to be met with in practice. Having thus shown the manner of finding such fluents as can be truly exhibited in algebraic terms, it remains now to say something with regard to those other forms of expressions involving one variable quantity only; which yet are so affected by compound divisors and radical quantities, that their fluents cannot be accurately determined by any method whatsoever. The only method with regard to these, of which there are innume In order to show the usefulness of fluxions, we shall give an example or two, 1. Suppose it were required to divide any given right line A B into two such parts, A C, C B, that their products, or rectangles may be the greatest possible. Let A B a, and let the part AC considered as variable (by the motion of C towards B) be denoted by x. Then BC being a-x, we have AC X BC ax-xx, whose fluxion ax - 2 x being put - 0, we get a i 2 x x; and, consequently, x = a. Hence it appears that AC (or x) must be exactly one half of AB. = suppose x to remain constant, and y, z, to vary; the product y, and consequentlyy z, will be the greatest when y = z. Or if y remain constant, the product x z, and consequently y rz, will be greatest when x = z. Thus it appears that the parts must be equal. And in like manner it may be shown, that whatever be the number of parts, they will be equal. Ex. 3. Given x + y + z = a, and x yì z3 a maximum, to find x, y, z. I As x, y, z, must have some certain determinate values to answer these conditions, let us suppose such a value of y to remain constant, whilst r and z vary till they answer the conditions, and then += 0 and 23 x + 3 x z3 i = 0; hence, 3 x zi 3 x ż =- -ż 23 2 i = 3 x. Now let us suppose the value of z to remain constant, and x and y to vary, so as to satisfy the conditions; then+y= = 0, y x + 2 x y y = 0; 2x y y 2 x i hence, i = — y y' tion, these values of y and z in terms of x, ..y=2x; substitute in the given equa. and x + 2 x + 3 x = a, or 6 x = a, 1 hence, x = a; . y 1 1 =རཱ; 2 6 like manner, whatever be the number of unknown quantities, make any one of them variable with each of the rest, and the values of each in terms of that one quantity will be obtained; and by substituting the values of each in terms of that one, in the given equation, you will get the value of that quantity, and thence the values of the others. Ex. 4. To inscribe the greatest parallelogram D F G I, in a given triangle A B C, fig. 10. Draw BH perpendicular to A C; put AC a, BH b, BE x, then E H and by similar triangles, b: a b x; ax hence, the area D F GI b max. or x x − x = = max. .. b x 2 x x b x hence, x = b; therefore E = 1 0: -B H. B Ex. 5. Let A B C represent a cone, A C the diameter of the base to in FLY, in zoology, a large order of insects, or rather an indeterminate appellation, used to express a vast variety of insects belonging to different orders. Entomologists apply the term only to individuals of the genus Musca. TOMOLOGY and Musca. See EN FLY, in mechanics, a cross with leaden weights at its ends, or rather a heavy wheel at right angles to the axis of a windlass, jack, or the like; by means of which the force of the power, whatever it be, is not only preserved, but equally distributed in all parts of the revolution of the machine. The fly may be applied to several sorts of engines, whether moved by men, horses, wind, or water, or any other animate or inanimate power; and is of great use in those parts of an engine which have a quick circular motion, and where the power of the resistance acts unequally in the different parts of a revolution. This has made some people imagine, that the fly adds a new power; but though it may be truly said to facilitate the motion, by making it more uniform, yet upon the whole it causes a loss of power, and not an increase for as the fly has no motion of its own, it certainly requires a constant force to keep it in motion; not to mention the friction of the pivots of the axis, and the resistance of the air. The reason, therefore, why the fly becomes useful in many engines, is not, that it adds a new force to them, but because, in cases where the power acts unequally, it serves as a moderator, to make the motion of revolution almost every where equal: for as the fly has accumulated in itself a great degree of power, which it equally and gradually exerts, and as equally and gradually receives, it makes the motion in all parts of the revolution pretty nearly equal and uniform. The consequence of this is, that the engine becomes more easy and convenient to be acted on and moved by the impelling force; and this is the only benefit obtained by the fly. The best form for a fly, is that of a heavy wheel or circle, of a fit size, as this will not only meet with less resistance from the air, but being continuous, and the weight every where equally distributed through the perimeter of the wheel, the motion will be more easy, uniform, and regular. In this form, the fly is most aptly applied to the perpendicular drill, which it likewise serves to keep upright by its centrifugal force: also to a wind. |