If the quantity Pp diminishes still farther, and becomes indefinitely small, the point m will almost coincide with M, and the secant become almost a tangent. But if e vanishes, the ratio already found is reduced to y, PS becomes PT, Sm becomes TM, which will P' , now have only the point M' in common with the curve, and the subtengent is determined by this limit. Thus if AMm is a parabola, we must substitute xte to x in the equation y=vpx, and we shall have y=p} (x+e)?=p mo + == x2 &c., which gives P=}v, whence PT = =2x, as we have & Wh P already found it. See Conic Sections. Here we must observe that the quantities Pp, or Mr, and rni, which gradually diminish as the point p approaches the point P, are the respective elements of the abscissa AP, and of the ordinate MP. These elements, however small we suppose them, preserve the same ratio to one another as the finite quantities to which they belong, as is visible by the mere inspection of the similar triangles TPM, and Mmr. As the finite quantities x and y have their fluxions x and ý, these fluxions have also in turn their flusions. These are called second fluxions, to distinguish them from the first, and are denoted by 8, 3. Similarly the third fluxions are marked i, , &c. RULES FOR FINDING THE FLUXIONS OF ANY PROPOSED FUNCTIONS. m 7. Given the equation y=ax, if we suppose that z 'receives any infinitely small increase, denoted by , y will also receive an increase, which we shall denote by y, and we shall have y+y = ar+ar; whence y=ar; and this is the fluxion of the proposed equation. If the equation had been b+y=-c, we should have found the same result'; for constant quantities have no fluxion. 8. And whenever the variable quantities do not exceed the first degree, we shall obtain the fluxions of the proposed quantities, by erasing the constant terms and substituting in place of the variable quantities their proper fluxions. If, for example, it were required to find the fluxion of bx+cy=”:+f, we should find by the abovo rule bi +cj =* But if the variable quantities are raised to other powers than the first; if, for example, we have y=cm, then supposing that a becomes m.m-1 *+* we shall have y+y=(x+x)" **+m2*13+ 2 &c. Now s?, 23, &c. vanish in respect to ~; consequently there remains y=mxml.z. Hence if m=2, y=2xr; if m=3, y=3r* :, &c. 9. In general, to find the fluxion of a variable quantity raised to any power, diminish its exponent by unity, and multiply the result by its original exponent and by its fluxion, 10. When two variables - and y are multiplied together, then flux (ry)=(x++) (y+) — xy=yč + xy + xy=y2 + xy, because my vanishes. Similarly flux (xyz) =z * finx (xy) + xyż=xyz +azy+yzi, and flux (uryz) =yz flux (u.x) + ut flur (yz) = uxyz + uxzy + wyze + sytu. Hence in general, to find the fluxion of the product of any number of variable quantities, find the flucions of one only at a time, as if all the rest were constant ; do the same thing with respect to cach quantity, and then collecl together all these fluxions. For example let the quantity be xy. If we consider y only to vary, we have my; and if only « be supposed to vary, we have By s*r. Therefore flux (x*y)=xy+3yxor. Similarly flux (x*y*%*) =3x*z*y*y + 4x*yox?2+2xyorič. 11. Let there now be proposed the fraction ; I write it thus, y 9 Aux m wi ху ух-ху yyy yy Hence to find the fluxion of a fraction containing variable quantities, we must 1°, multiply the denominator by the fluxion of the numeralor ; 2dly, multiply the numeralor by the fuxion of the denominator ; 3dly, subtract the latter product from the former, and divide the remainder by the square of the denominator. 1 and applying the rule at art. (9) we shall find 1 2(a +2x) (az +z")" (az +2zz)= man (az+z) Consequently to find the flusion of any radical of the mth degree, we must divide the fluxion of the quantity beneath the radical sign by the exponent m, and by the mth root of this quantity raised to the power m--1. By these few rules, we are enabled to find the fluxion of any algebraic quantity whatever. Examples for Practice. 1. Required the fluxion of Ans: y yy of v(ay+yy) 2. Ans (1 9+y) y (9y+yy) 3. of (ax + brx + cruja The square xy OF SECOND, THIRD, &c. FLUXIONS. 13. The second Auxion of a quantity is the fluxion of the first fluxion. The third fluxion is the fluxion of the second, and so on : thus ä signifies the fluxion of 's ; C, that of ö. of the fluxion 2 is written **; its mth power is written zi, &c. From what we have just said respecting first Auxions, it will be easy to find the second fluxions, &c. Thus if we require the second fluxions of x*; we shall have for the first fluxion 2x2; consequently the second Auxion will be 2*x+2x8=2x2+2x.f. Similarly, since flux (cm)=mxm..., we shall 2d-flux (***)=mmmhö+m.m-ls*** again flux (xy)=xy+yx'; therefore 2d flux (xy)=xi+yö +2yr. Since flur yx —xy. we easily infer that 2d four yy y yy ở T_T 2 x ty_2ry yyyy yy Y yy yy 2xy* +y*x -— xyy—2yxy, and in the same manner with other func ys tions. By the same principles we are enabled to find the third, fourth, &c. Auxions, and in general the fluxions of any degree, of all sorts of quantities affected with and y. For example the fluxion of yż is yä täyt.... that of ✓ (**+y*) that of yd is + yö_yvin and that of the v (x2+y) y y infinitely great quantity ? is &c. &c. y + + **+yy; is ar 14. In order to shorten the calculation of the second fluxions of several variable quantities, we generally suppose one of the Arst 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. Forexample, if we required the fluxion of Yx; on the supposition that ; is constant, we should find the required Auxion to be ; _yty ye and if we considered y as constant, we should obtain : yx. + 15. Hitherto we have supposed that the variable quantities of which the fuxions were required, increased all at the same time. If some increased, while others diminished, this would cause no dif. ficulty, for I and y may be positive or negative, as well as any other algebraic quantities. OF LOGARITHMIC AND EXPONENTIAL FLUXIONS. 3.23 16. Let it be required to find the fluxion of the natural logarithm of the variable r. We shall denote this logarithm by lx, and making lx=x, we have z+z=f(x+*!; this gives z or flux (lx)=1 2.3 (1+i) - lx=l(1+ + &c = neglecting the higher powers of c. Hence the fluxion of the logarithm of any quantity is equal to the furion of thal quantity divided by itself. Consequently in a system in which the modulus=m, we shall have flux (lx),= But in the sabsequent articles we shall treat only of the natural or hyperbolic logarithms whose modulus=1. By the above rule, we easily find Aux (lx") = 1.29-1 NX &c. |