which is therefore equal to the fluxion of the quantity, divided by the quantity itself. Hence the fluxion of the byp. log. 24. By means of the fluxions of logarithms, are usually determined those of exponential quantities, that is, quantities which have their exponent a flowing or variable letter. These exponentials are of two kinds, namely, when the root is a constant quantity, as e, and when the root is variable as well as the exponent, as y*. 25. In the first case put the exponential, whose fluxion is to be found, equal to a single variable quantity z, namely, z=ex; then take the logarithm of each, so shall log. z = x X log. e; take the fluxions of these, so shall X log. e, by the last : 2 article hence ż=zx Xlog. e=exxlog. e, which is the fluxion of the proposed quantity e or 2; and which therefore is equal to the said given quantity drawn into the fluxion of the exponent, and into the log. of the root. Hence also, the fluxion of (a+c)nx is (a+c)no × nx × log. (a+c). 26. In like manner, in the second case, put the given quan. tity yz; then the logarithms give log. z=x Xlog. y, and the fluxions give = x×log. y+z; henceż zxxlog. y+ zxy y y = (by substituting y for z) y xxlog. y+xy-1y, which is the fluxion of the proposed quantity y; and which therefore consists of two terms, of which 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. VOL. II. 41 OF 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 highest orders, as second, third, fourth, &c. fluxions. But 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. 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,ï, ï, &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 art. 8, page 306; where, if a denote the absciss AP, and y the ordinate PQ and if the ordinate PQ or y flow along the absciss AP or x, with a uniform motion; then the fluxion of x, namely, pp or ar, is a constant quantity, or =0, in all the figures. Also, in fig. 1, in which AQ is a right line, y=rq, or the fluxion of PQ, is a constant quantity, or y 0; for, the angle q, the angle A, being constant, or is to rq, or to y, in a constant ratio. But in the 2d fig. rq, or the fluxion of p, continually increases more and more; and 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 x is every where equal to a2y; then, taking the fluxions it is a2 y=3x2; 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 a3 y=3x2 x, the 2d fluxions a2 y=6xx2, 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 3y2y, con sidered as the rectangle of 3y3y2ÿ+6yy: B 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 xy, 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-y, the fluxion of xy will in that case be xy-xy. This may be illustrated by the annexed rectangle, APQR = xy, supposed to be generated by the motion of the line PQ from A towards c, and by the motion of the line вQ from B towards A: For, by the motion of PQ, from A towards c, the rectangle is increased, and its fluxion is + xy ; but by the motion of RQ, from в towards A, the rectangle is decreased, and the fluxion of the decrease is ry; therefore, taking the fluxion of the decrease from that of the increase, the fluxion of the rectangle xy, when x increases and y decreases, is xy -xy. R ४ OC الا A P REMARK BY THE EDITOR. The fluxion of the algebraic quantity xy is properly yx+xy We should always use in all cases of increase or decrease. the signs of the fluxions of algebraic expressions as those signs arise from the known rules, without considering whether the quantities increase or decrease; but in denoting, algebraically, the simple fluxions of geometrical quantities, we should prefix the sign minus to the fluxions of such a decrease; and thus we may, in any case, use the fluxions of algebraic equations, together with the fluxions derived from geometrical figures, without embarrassment or apprehension of error. 30. We 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 flucion 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 0-43429443. 7th, For the fluxion of an Exponential quantity having the Root Constant. Multiply altogether, 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, and 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 |