fore, taking the fluxion of the decrease from that of the increase, the fluxion of the rectangle ry, when x increases and y decreases, is xy — xj. 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 Ist 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 bryz is 3. The fluxion of cx x (ax- cy) is · 4. The fluxion of ry" is 5. The fluxion of xyz" is 6. The fluxion of (x + y) × (x − y) is 7. The fluxion of 2ax2 is 8. The fluxion of 2x3 is 9. The fluxion of 3x^y is 22. The fluxion of ✔ (a2 + x2) or (a2 + x2)3 is 23. The fluxion of √(a2 — x2) or (a2 → x2) 24. The fluxion of √(2rx — xx) or (2rx 25. The fluxion of 1 (a2 √(x2-x2) or (a• — 2") 26. The fluxion of (ax − xx)3 is is xx)ž is is 27. The 42. The fluxion of the hyp. log. of ar is is 48. The 62. The second fluxion of ry, when i is constant, is 63. The second fluxion of an is 64. The third fluxion of ", when x is constant, is 65. The third fluxion of xy is THE INVERSE METHOD, OR THE FINDING OF FLUENTS. 32. It has been observed, that a Fluent, or Flowing Quantity, is the variable quantity which is considered as increasing or decreasing. Or, the fluent of a given fluxion, is such a quantity, that its fluxion, found according to the foregoing rules, shall be the same as the fluxion given or proposed. 33. It may further be observed, that Contemporary Fluents, or Contemporary Fluxions, are such as flow together, or for the same time. When contemporary fluents are always equal, or in any constant ratio; then also are their fluxions respectively either equal, or in that same constant ratio. That is, if xy, then is xy; or if xy::n: 1, then is jn: 1; or if any, then is x = nj. 34. It is easy to find the fluxions to all the given forms of fluents; but, on the contrary, it is difficult to find the fluents of many given fluxions; and indeed there are numberless cases in which this cannot at all be done, excepting by the quadrature and rectification of curve lines, or by logarithms, or by infinite series. For, it is only in certain particular forms and cases that the fluents of given fluxions can be found; there being no method of performing this universally, a priori, by a direct investigation, like finding the fluxion of a given fluent quantity. We can only therefore lay down a few rules for such forms of fluxions as we know, from the direct method, belong to such and such kinds of flowing quantities: and these rules, it is evident, must chiefly consist in performing such operations as are the reverse of those by which the fluxions are found of given fluent quantities. The principal cases of which are as follow. 35. To find the Fluent of a Simple Fluxion; or of that in which there is no variable quantity, and only one fluxional quantity. This is done by barely substituting the variable or flowing quantity instead of its fluxion; being the result or reverse of the notation only.—Thus, The fluent of ax is ax. The fluent of aỷ + 2ỷ is ay + 2y. The fluent of √√ a2 + x2 is √ a2 + x2. 36. When any Power of a flowing quantity is Multiplied by the Fluxion of the Root: Then, having substituted, as before, the flowing quantity, for its fluxion, divide the result by the new index of the power. Or, which is the same thing, take out, or divide by, the fluxion of the root; add 1 to the index of the power; and divide by the index so increased. Which is the reverse of the 1st rule for finding fluxions. So, if the fluxion proposed be Leave out, or divide by, x, then it is add 1 to the index, and it is divide by the index 6, and it is 3.xx. 3x or 16, which is the fluent of the proposed fluxion 3.xx. In like manner, The fluent of 2axx is ax2. The fluent of 3x2x is x3. The |