y y2 that is, F.= If the numerator of the fraction this for 29-1 P be a constant quantity, having its fluxion equal to 0, the fluxion of XP = substitute such fraction is equal to the flux- we have p xPxqx ¶x (2—1)— i. ion of the denominator drawn into the numerator, and divided by the square of the denominator. Whence it appears, that the same rule has place both for inte gral and fractional indices, and for negative consequently also ones; that is, 1x F.-=— nx―n— For x-n= and F. 1x That is, F. an nxn — 1 x. that the number of terms in that F. nnxn-ix. To find the fluxion of a quantity having a fractional index. Multiply the proposed quantity by the index, reduce that index by unity, and multiply by the fluxion of the simple quantity; that is, F. annan → 12, whether n be integral or fractional. We have demonstrated this above, in the case where n is an integer, and have therefore in this place only to attend to the case in which n is fractional. Let, therefore, n= and make P 180 == 1 1 bolic logarithm of a tity by the quantity itself, and it x where Therefore the fluxion of any log. | the formula for the sines of sums. of arcs, we have sin. (x + x)=sin. of x, is equal to M X z cos. zsin. x cos. z, the radius M is the modulus of the system. being assumed equal to unity. But If it be the common, or Brigg's lo- the sine of an are indefinitely garithm, then M = 0 43429, &c. and small does not differ sensibly from for the hyperbola logarithm,M=1. that are itself, nor its cosine differ To find the fluxion of exponen- perceptibly from radius; hence we tial quantities, that is, quantities have sin. zz, and cos. z = 1; and which have their exponent a flow-therefore sin. (z + x) = sin. ≈ + 2 ing or variable letter. These are of two kinds, viz. when the root is a constant quantity, as ex; and when the root is variable, as yx. cos. z; whence sin. (z+z) — sin. z, or (sin. z) =ż cos. z, viz. the fluxion of the sine of an arc whose radius is unity, is equal to the pro into the cosine of the same arc. In the former case, put the product of the fluxion of the angle posed exponential exz, a single variable quantity; then take the logarithm of each, so shall log. z = X log. e; take the fluxions of these, so shall In like manner the fluxion of cos. z, or cos. (z + z) — cos. %= Cos. % COS. 2- sin. z sin. z -COS. z, or since cos. (* + *)=cos. ≈ cos. sin. sin; therefore, be. cause sin. % %, and cos. % = 1, we have F. cos. z=cos. zz sin. z %% =% X F. sin. mz = (by substituting yo for :) g ở X log. 3 + xyx1 g, is the fluxion of the proposed exponential yx; which therefore consists of two terms, of which the one is the fluxion of the proposed quantity,considering the exponent only as constant, and the other is the fluxion of the same quantity, considering the root as constant. To find the fluxions of sines, cosines, &c. Suppose we require the fluxion of the sine of the angle or arc de noted by z, we must suppose that by a motion of one of the legs including the angle, it becomes + %, then sin. (z + z) — sin. z is the fluxion of sin, z. But according to therefore they may be found, in the same manner, by the general rules already delivered. Thus, by the third rule, the first fluxion of x3 is 3 x2 x, and if x be supposed constant, cr if the root & be generated with an equable celerity, the fluxion of 3 x2 x, or 3 x x x2, will be 3x, x 2 xx2 = 6 xx2, which is the second fluxion of x3, and 6 28 will be its third fluxion; but if the celerity with which a is generated be variable, either increasing or decreasing, then a being variable, will have its fluxion denoted by x, &c. In this case the fluxion of 3 x2xx will be, by the first and third rules, 6 x x × x + 3x2 × x = 6 x x2 + 3x2x, the second fluxion of rs. And the third fluxion of a obtained in like manner from the last, will be 6 × x2 + 6x × 2xx +6 x x × ï + 3 x2 = 623 ... then 2 2xy + yx, &c. If the function proposed were axn, we should find F. an nax-x; the factors na and being regarded as constant in the first fluxion nax-1, to obtain the second fluxion it will suffice to make an flow, and to multiply the result by na x; but F. xn−1 = (n-1)x-2x; we have, therefore, 2nd F. axnn (n-1) axn—2x2. 4th F. axnn (n−1) (n−2) (n−3) mth F. axn=n (n−1) n—2) • (n-m + 1) an―m xm, nth F. a xn−n (n−1) (n−2) · 3.2.1. aan in which state it admits no longer of being put into fluxions, as it contains no variable quantity, or, in other words, its fluxion is equal to o. Inverse Method.-In the direct method of fluxions our object is to find the fluxion of any flowing or variable quantity, for which the proper rules have been laid down in the preceding part of this arti cle, and the operation is therefore always direct and easily accom where it is required to find the plished; but in the inverse method, fluents of given fluxions, the ope ration is much more difficult, as no rules can be laid down that will be sufficient for performing this in all cases; for though every flowing quantity has its peculiar fluxion, yet every fluxion has not its fluent, at least not without having recourse to infinite series, quadrature of curves, or other methods of approximation. There are, however, a few rules which may be useful. 1. When any power or root of the variable quantity is multiplied by the fluxion of that quantity. Substitute the variable quantity instead of its flaxion, which will increase the index of the power or root by unity; then divide the quantity by the index thus increased, and it will be the fluent required. That is, 2. When the root under the vinculum is a compound tity, and the index of the part or factor without the vinculum, increased by unity, is some multiple of that under the vinculum. m being supposed not to exceed n, for it is manifest that in the case of n being integral, the function Put a single variable quantity aan has only a limited number of for the compound root, and subfluxions, of which the most eleva-stitute its powers and fluxions inted is in the nth, and which of stead of those of the quantity itcourse is expressed by the for- self. Find the fluent of this simple mula, fluxion, and then re-establish the 182 of x as also y, y, y, &c. in continual proportion. Then divide the square of the given fluxional expression by the second fluxion, just found, and the quotient will in many cases be the fluent. Or the same rule may be delivered thus: In the given fluxion write a for x, y for y, &c. and call the result G, taking also the fluxion of this quantity &; then make &: FG: F, so shall the fourth proportional F be the fluent, as before. be the true fluent, by taking the It may be proved whether this fluxion of it again. If it agree with the proposed fluxion, it will show that the fluent is right; otherwise or -1x, is the hyp. log. of a; the method fails. Thus, if it be proposed to find is 2x hyp. log. of x or the fluent of nan-1 x. Here F hyp. log. of x2; nan-x; write first a for x, and it is nan x, or nxn = G; the is the hyp. log. of a+x; fluxion of this is Ġ=n2 1 x; therefore G: F =G: F becomes is the hyp. log. of a + n2 xn-1 x : nxn−1 x = nxn ; xn = F, the fluent sought. of a + x' of a+ To find Fluents by Series. is hyp. log. of (x+any form not included in the preWhen the proposed fluxion is of ceding rules, nor given in the annexed table, there is no other method of obtaining the fluent but by is hyp. log. (x+a+a series, at least it will generally be found most convenient and direct, and will apply in all cases; the rule is as follows: √x2+2 ax); hyp. log. of "+x a-x ; is hyp. log. of a a2 + xx a + √ a2±x2 Expand the radical or fraction into a series by the binomial theoorem, and multiply the several terms by the fluxion of the variable, then take the fluents of the several terms, which will be the fluent required. It should be observed, however, These points are thus dènominated from a remarkable property that obtains in all the three figures; viz. that if lines be drawn from the two foci, to any point in the curve, and a tangent be drawn at their point of meeting, then those two lines will form equal angles with the tangent. And since it is a known property in optics, that the angle of incidence is equal to the angle of reflection; therefore, all rays issuing from the one focus, and falling on the various points of the curve, will be reflected into the other focus, or into the line Now taking the fluents of each directed to the other focus, which term, we have, 6.x7 -&c. 875a6 15a2 125a4 ·} And the method is the same in all similar cases. If the index of the proposed fluxion be fractional the series will be infinite; but when it is integral the series will terminate, and the fluent will be finite. Thus it will be found that, + xm x xm axm-ı x + a = is the case in the hyperbola, whence the denominated foci or burning points. Properties relating to the foci of the conic sections may be enumerated as follows: 1. In the ellipse and parabola sum of any two lines drawn from the transverse axis is equal to the the two foci, to meet in any point of the curve. bola the transverse axis is equal to And in the hyperthe difference of the same lines. 2. Again, the square of the dis tance between the foci is equal to +, &c. + the difference of the squares of the two axis, in the ellipse and para bola, but in the hyperbola it is equal to their sum. ·+, &c. ± am h. log. (x + a) m being any integer number. FLY, in Mechanics, is a heavy weight applied to some part of a machine, principally in order to render its motion uniform, though it is sometimes employed for the purpose of increasing the effect, as in the coining engine. It regulates the motion, because its momentum is not easily disturbed. FOCAL Distance, in the Conic Sections, is the distance of the focus from some fixed point; viz. from the vertex of the parabola, and from the centre in the ellipse and hyperbola. FOCUS, is that point in the transverse axis of a conic section, at which the double ordinate is equal to the parameter, or to a third proportional to the transverse and conjugate axis. 3. In the parabola the focal distance is equal to the parameter, or of the ordinate at the focus. 4. Also a line drawn from the focus to any point of the curve is equal to the focal distance plus the ordinate at that point. Focus, in Optics, is a point wherein several rays concur, or are collected, after having undergone either refraction or reflection. This point is thus denominated, because the rays being here brought together and united, their joint effect is sufficient to burn bodies exposed to their action; and hence this point is called the focus, or burning point. It must be observed, however, that the focus is not, strictly speaking, a point; for the rays are not accurately collected into one and the same place or point; owing to the different nature and refrangibility |