the one is the fluxion of the given quantity considering the exponent as constant, and the other the fluxion of the same quantity considering the root as constant. OF SECOND, THIRD, &c. FLUXIONS. HAVING explained the manner of considering and determining the first fluxions of, flowing or variable quantities; it remains now to consider those of the higher orders, as second, third, fourth, &c. fluxions. 27. If the rate or celerity with which any flowing quantity changes its magnitude, be constant, or the same at every position; then is the fluxion of it also constantly the same. But if the variation of magnitude be continually changing, either increasing or decreasing; then will there be a certain degree of fluxion peculiar to every point or position; and the rate of variation or change in the fluxion, is called the Fluxion of the Fluxion, or the Second Fluxion of the given fluent quantity. In like manner, the variation or fluxion of this second fluxion, is called the Third Fluxion of the first proposed fluent quantity; and so on. These orders of fluxions are denoted by the same fluent letter, with the corresponding number of points over it: namely, two points for the second fluxion, three points for the third fluxion, four points for the fourth fluxion, and so on. So, the different orders of the fluxion of x, are x, x, x, x, &c; where each is the fluxion of the one next before it. :. :: 28. This description of the higher orders of fluxions may be illustrated by the figures exhibited in page 277; where, if r denote the absciss AP, and Y the ordinate PQ; and if the ordinate po or y flow along the absciss AP or x, with a uniform motion; then the fuxion of x, namely, Pp or or, is a constant quantity, or x = 0, in all the figures. Also, in fig. 1, in which aQ is a right line,j = rq, or the fluxion of PQ, is a constant quantity, or y = 0; for, the angle o, the angle A, being constant, or is to rq, or to j, in a constant ratio. But in the 2d fig. rq, or the Auxion of pa, continually increases more and more; and 286 OF SECOND, THIRD, &c. FLUXIONS. in fig. 3 it continually decreases more and more, and therefore in both these cases y has a second fluxion, being positive in fig. 2, but negative in fig. 3. And so on, for the other orders of fluxions. = Thus if, for instance, the nature of the curve be such, that r3 is everywhere equal to y; then, taking the fluxions, it is a'j 3xx; and, considering & always as a constant quantity, and taking always the fluxions, the equations of the several orders of fluxions will be as below, viz. the 1st fluxions a2 = 3x2x. :: the 4th fluxions ay = 0, and all the higher fluxions also = 0, or nothing. Also, the higher orders of fluxions are found in the same manner as the lower ones, Thus, the first fluxion of y3 is its 2d flux. or the flux. of 3y', considered as the rectangle of 31, and j, is 3y'j; and the flux. of this again, or the 3d3y+18yÿÿ + 6ÿ3. flux. of y3, is 29. In the foregoing articles, it has been supposed that the fluents increase, or that their fluxions are positive; but it often happens that some fluents decrease, and that therefore their fluxions are negative: and whenever this is the case, the sign of the fluxion must be changed, or made contrary to that of the fluent. So, of the rectangle ry, when both x and y increase together, the fluxion is xy + xy; but if one of them, as y, decrease, while the other, x, increases; then, the fluxion of y being, the fluxion of xy will in that case be xy-xỷ. This may be illustrated by the annexed rectangle, APQR = ry, supposed to be generated by the motion of the line pafrom A towards c, and by the motion of the line RQ from B towards A: For, by the motion of pa, from A towards c, the rectangle is increased, and its fluxion is xy; but, by the motion of RQ, from B towards A, the rectangle is decreased, and the fluxion of the decrease is x; B R PC therefore, therefore, taking the fluxion of the decrease from that of the increase, the fluxion of the rectangle ry, when a increases and y decreases, is xy - xỷ. 30. We may now collect all the rules together, which have been demonstrated in the foregoing articles, for finding the fluxions of all sorts of quantities. And hence, 1st, For the fluxion of Any Power of a flowing quantity. -Multiply all together the exponent of the power, the fluxion of the root, and the power next less by 1 of the same root. 2d, For the fluxion of the Rectangle of two quantities.Multiply each quantity by the fluxion of the other, and connect the two products together by their proper signs. 3d, For the fluxion of the Continual Product of any number of flowing quantities.-Multiply the fluxion of each quantity by the product of all the other quantities, and connect all the products together by their proper signs. 4th, For the fluxion of a Fraction.-From the fluxion of the numerator drawn into the denominator, subtract the fluxion of the denominator drawn into the numerator, and divide the result by the square of the denominator. 5th, Or, the 2d, 3d, and 4th cases may be all included under one, and performed thus.-Take the fluxion of the given expression as often as there are variable quantities in it, supposing first only one of them variable, and the rest constant; then another variable, and the rest constant; and so on, till they have all in their turns been singly supposed variable; and connect all these fluxions together with their own signs. 6th, For the fluxion of a Logarithm.-Divide the fluxion of the quantity by the quantity itself, and multiply the result by the modulus of the system of logarithms. Note. The modulus of the hyperbolic logarithms is 1, and the modulus of the common logs, is 0.43429448. 7th, For the fluxion of an Exponential quantity, having the Root Constant.-Multiply all together, the given quantity the fluxion of its exponent, and the hyp. log. of the root. 8th, For the fluxion of an Exponential quantity having the Root Variable.-To the fluxion of the given quantity, found by the 1st rule, as if the root only were variable, add the fluxion of the same quantity found by the 7th rule, as if the exponent only were variable; and the sum will be the fluxion for both of them variable. Note. When the given quantity consists of several terms, find the fluxion of each term separately, and connect them all together with their proper signs. 31. PRACTICAL 31. PRACTICAL EXAMPLES TO EXERCISE THE FOREGOING RULES. 1. The fluxion of ary is 2. The fluxion of brys is 3. The fluxion of cx x (ax 4. The fluxion of my" is 5. The fluxion of xy" is cy) is 6. The fluxion of (x + y) × (x − y) is 7. The fluxion of 2ax2 is 8. The fluxion of 2.r3 is 9. The fluxion of 3rty is 10. The fluxion of 4x3y+ is 11. The fluxion of ar2y — xžy3 is 27. The fluxion of 2x/a2± x2 is 28. The fluxion of (a2 — x2)* is I 29. The fluxion of √xz or (xz)2 is 30. The fluxion of /xzzz or (xz – zz) is 42. The fluxion of the hyp. log. of ar is 43. The fluxion of the hyp. log. of 1+x is 44. The fluxion of the hyp. log. of 1 x is 45. The fluxion of the hyp. log. of x2 is 46. The fluxion of the hyp. log. of √z is. 47. The fluxion of the hyp. log. of rm is VOL. II. U 48. The |