OF SECOND, THIRD, &c. FLUXIONS. .. 13. The second fluxion of a quantity is the fluxion of the first fluxion. The third fluxion is the fluxion of the second, and so on: thus a signifies the fluxion of x; x, that of x. The square of the fluxion is written ; its mth power is written ", &c. From what we have just said respecting first fluxions, it will be easy to find the second fluxions, &c. Thus if we require the second fluxions of r2; we shall have for the first fluxion 2xx; consequently the second fluxion will be 2xx+2xx=2x2+2xx. Similarly, since flux (x)=mxm-1x, we shall 2d-flux (xTM)=mxTM-1 x+m.m-1x=22 again flux (xy) = xỷ+yx; therefore 2d flux (xy)=xy+yx+2yx. Since flur x ___yx—xy__ xy, we easily infer that 2d flur () y yy y yy Qxya + y2x — xyÿ—2yxy, and in the same manner with other func tions. y3 By the same principles we are enabled to find the third, fourth, &c. fluxions, and in general the fluxions of any degree, of all sorts of quantities affected with x and y. For example the fluxion of yr is y+y....that of √ (¿1+y') 14. In order to shorten the calculation of the second fluxions of several variable quantities, we generally suppose one of the first fluxions to be constant; that is, we refer the other fluxions to that one, as to a fixed term of comparison. We shall soon see examples of this process. This kind of supposition simplifies the labour by causing all those terms to disappear which are affected with the fluxion of the quantity assumed as constant. For example, if we required the fluxion of; on the supposition that is constant, we should find the required fluxion to be ryzy and if we considered y as constant, we should obtain + yx. y 15. Hitherto we have supposed that the variable quantities of which the fluxions were required, increased all at the same time. If some increased, while others diminished, this would cause no difficulty, for x and y may be positive or negative, as well as any other algebraic quantities. OF LOGARITHMIC AND EXPONENTIAL FLUXIONS. 16. Let it be required to find the fluxion of the natural logarithm of the variable x. We shall denote this logarithm by lx, and making lx=x, we have += (x+x); this gives z or flux (x)=l (x + x) — lx = l(1 + =)== the higher powers of x. Hence the fluxion of the logarithm of any quantity is equal to the fluxion of thal quantity divided by itself. Consequently in a system mx But in in which the modulus-m, we shall have flux (lx)= x the subsequent articles we shall treat only of the natural or hyperbolic logarithms whose modulus=1. By the above rule, we easily find flux (lx") = #c---- Nx &c. 17. If we had the powers of logarithms, or even logarithms of logarithms, it would be easy to find their fluxions. For example, let y=(lr)”, we shall have y=m (la)=-1= If we had y=x′′ (kz)", we should obtain y=mxm1 x (lx)" + nxm-1 x (lx)"1 = xTM1 x (lx)~1 (n+mlx), &c. Again let y=llr; make r=z, and we shall have ==, ¿ =» 18. The equation flux ((x) === gives = flux (le). Consequently the fluxion of any quantity is equal to the product of that quantity by the the fluxion of its logarithm. This rule may be applied to facilitate finding the fluxions of quantities of any sort, even if algebraic. For example flux (x)=x flux lxTM= mxml'x. x y Flux (zy)=xy( +)=y+y. Flux ( yx + xy mx x The same principle may be applied with success in finding the fluxions of exponential quantities; by which name is to be understood, such quantities as have variable exponents. Such are a*, **, &c. which are of the first order; 2, which is of the second, &c. The fluxion of a*, will be according to this rule a* flux la*=a* ffux (xla) = a*xla. Therefore if e be the number 2.7182818, of which the logarithm is 1, we shall have flux (e") = ex. Similarly flux (x3) = xo flux (ylx) = x} (ylx + Ya), &c. 19. We might also have found these fluxions in the following We have seen that n = 1 + In + (ln)2 (ln)3 + + &c. manner. 2 2.3 Suppose now that n=a*, and substitute this value in place of ʼn, and 2.3 we shall find a′′=1+la”+ (la), (la)3 Now la*=xla, and 2 (la) = (xla)2 = xl'a; therefore a*—1+xla+ and consequently flux (a*) = xla + xx l'a + With respect to such exponentials as, their fluxion is easily found: for we have flux (x'")=xo"flux (y' lx) = xo" {y =+ylx (zły + zy =y +ZY lx + xlxly). If x=y=e, we shall have eez for the I y fluxion of e'. In a similar manner we may find the second, third, &c. fluxions of logarithmic and exponential quantities, but it is needless to dwell longer on this head. OF THE FLUXIONS OF SINES, COSINES, &c. AND OTHER CIRCULAR FUNCTIONS. 20. Let sin x=y, we shall have y+y=sin(x+x) = sin x cos z + sin x cos x. Now x representing an infinitely small arc, we shall have 1st, cos x=1; 2dly, sin x=x. Therefore y+y=sin + cos ■, or y= flux sin xx cos x. Consequently the fluxion of the sine of any arc is equal to the fluxion of that arc multiplied by its cosine. 21. Since flux sin x cos x, if we make x = 90°-y we shall have a-y, and flux cos y y sin y, a formula which we might have obtained in either of the two following manners. Ist. Sin2 + cos2 = 1. Therefore sin x (flux sin x) + cos x (flux cos x)=0, and flux cos x = (flux sin x) =— - sin x or 2dly, flux cos x =cos (x+x) x sin x -cos x= r sin r. - sin r cos = cos x cos x sin Hence the fluxion of the cosine of any arc is equal to the negative fluxion of that arc multiplied by its sine. = flux tang x. COS 2x cos 2x + sin 2 cos 2x Hence the fluxion of the tangent of any arc is equal to the fluxion of that arc divided by the square of its cosine. If instead of supposing the radius = 1, we had supposed it = a, 23. Let x=90°-y, we shall have flux cot y = sin y sin' y sin y y cot y. sin y These same formula may also be found in the following manner. Let the arc AB be denoted by z; its = E DA CD, =y. The similar triangles CBD, Bmr, give Similarly cos AB or = √(1-y'), and therefore or flux cos y ——¿ sin z. Let AT or tan z=t, we shall have t:1::y:√(1-y1), and con i (1—y1) Hence it follows that flux (4), or —— or flux cot 1 —— y'√(1—y3) as before. These rules enable us to find the first, second, &c. fluxions of any quantity in which there enter sines, cosines, &c. Thus, for Example. Flux (sin x)” = m (sin x)-1 x cos x-m sin x x cot x. Second flux (sin x) = x cos x — Second flux (cos x) 1 - sin x. = flux (— x sin x) — — x sin a—¿a cos . Flux (sin mx)= mx cos mx. |