that is constant, we should find the required fluxion to be yay ; y and if we considered y as constant, we should obtain * + *. 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 la, and making lux, we have z+z=l (x+x); this gives z or flux (lr)=l 3 2x2 3x3 - &c. =, neglecting the higher powers of x. Hence the fluxion of the logarithm of any quantity is equal to the flurion of that quantity divided by itself. Consequently in a system in which the modulus=m, we shall have flux (lx)= mx But in 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") = nx ,&c, Ans. 6pn1 m a+bx" Ans.(1+2) 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=(lt)", we shall have y=m (lx)m-1 If we had y=x (lz)", we should obtain y=mx1 x (lx)" + nxm-1 x (lx) -1 = xml x (lr) (n+mlx), &c. Again let y=llr; make lr=z, and we shall have 18. The equation flux (x) = gives = x flux (le). Conse quently 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 mxx if algebraic. For example flux (x) = x flux lem y y 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", x, &c. which are of the first order; r, which is of the second, &c. The fluxion of a", will be according to this rule a flux la=a ffux (xla) = axla. Therefore if e be the number 2.7182818, of which the logarithm is 1, we shall have flux (e) = ex. Similarly flux (xy) = x flux (ylx) = xy (jlx +), &c. 19. We might also have found these fluxions in the following manner. 2 We have seen that n = 1 + ln + (ln) (ln)3 + + &c. 2 2.3 Suppose now that n=a", and substitute this value in place of n, and we shall find a2=1+la2 + (la*)2 + (la)3 + &c. Now la=xla, and xlax3Ba + 2.3 xx Ba + &c. = xla 2 2.3 (la) = (xla) = x2la; therefore a=1+xla+ and consequently flux (a2) = xla + xx la + {1+ala+a+za+ &c. } = ila 2 2.3 2 2 + &c. With respect to such exponentials as, their fluxion is easily found: for we have (+ 12 + zlaly). If x=y=e, we shall have cez for the 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 cos + sin cos x. Now a representing an infinitely small arc, we shall have 1st, cos x=1; 2dly, sin x=x. Therefore y+y=sinx + cos z, or y= flux sin x=x cos x. Consequently the fluxion of the sine of any arc is equal to the fluxion of that are multiplied by its cosine. 21. Since flux sin x=x cos x, if we make x = 90° -y we shall have x - y, and flux cos y = y sin y, a formula which we might have obtained in either of the two following manners. 1st. Sin x + cosx = 1. Therefore sin x (flux sin x) + cos x sin (flux cos x) =0, and flux cos x = - (flux sin x) = -x sin x COS x (x+x) - COS COS x cos x sin Hence the fluxion of the cosine of fluxion of that arc multiplied by its sin (flux sin x) - flux COS 2 = = flux tang z. Hence the fluxion of the tangent of any 2 cos x 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, we should have had flux tang x = aa r x cos 'r 23. Let x=90°-y, we shall have flux cot y =; similarly sin 'y 1-_-flux sin y-y cos y sin y sin' y sin 'y y tang 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 cosine CD by z; its sine BD by y; and suppose the radius to be unity. Conceive now AB to be lengthened by the indefinitely small arc Bm, and draw mr perpendicularly on BG; we shall have Bm = 2, Br = -, and mr =y. The similar triangles CBD, Bmr, give or, G BT C DA Bm: BC:: Br: BD :: mr : CD, :: 1 :: - :(1-x2):: y: √(1-y) Similarly cos AB or x = √(1-9), and therefore & or flux cos : =-yx y =-: sin z. Let AT or tan z=t, we shall have t:1::y: √(1-y), and con y(1-y) Hence it follows that fluz (7), or = 1 X y y√(1-y2) sin z 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 & cos xm sin xTM & cot x. Second flux (sin x) = x cos 2 - 2 sin z. Second flux (cos x) = flux (- i sin x) = Flux (sin mx) = mx cos mx. Flux (cos mx) - mx sin mz. sin - cos z. Flux (sin x cos x) = cos - sin z = & cos 2x. (1+cos x) Since ✓ =cos x, we have flux ( 2 =-sin. 1+cos Similarly we shall find that flux (cos log x) = - flux log x sin log = = - sin log ; and that flux (2 sin 2)= sin x + zi cos 2. - flux cos x sin x -flux cot x sin s APPLICATION OF FLUXIONS TO THE THEORY OF CURVES. 25. Of all the problems that can be proposed respecting a curve, the most simple is that which requires us to draw a tangent to any point of the curve. Suppose the curve to be AM, its axis, AP; its coordinates AP and PM; it is evident that to draw a tangent to the point M, we have only to determine the subtangent PT. M B a Let us imagine the arc Mm to be infinitely small, draw the ordinate mp, infi nitely near to AP, and sup- T pose Mr parallel to Pp. Let, as usual, AP=r, PM=y, and we shall have Pp, or Mr-x, mr=y; and by similar triangles Mrm, TPM we have mr :M::MP: PT ory: *::y: PT= Consequently we have only to throw the equation of the curve into fluxions, in order to obtain the value of ✗ , and then to substitute this value in the y formula for the subtangent just found, and PT will be determined. 26. The expression for the tangent MT is ✓(+); that of the subnormal PN is PT' MN =√(*+*); and if through the point A we draw the line |