42. The fluxion of the hyp. log. of ar is 48. The 53. The fluxion of (a + c) is 54. The fluxion of 100 is 55. The fluxion of r2 is 61. The second fluxion of xy is 62. The second fluxion of xy, when is constant, is 63. The second fluxion of x" is 64. The third fluxion of rh, when 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 Quan tity, 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 x =; or if xy::n: 1, then is jn1; or if rny, then is ny. 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 a + 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 add 1 to the index, and it is divide by the index 6, and it is 3x5x. which is the fluent of the proposed fluxion 3xx. In like manner, The fluent of 2axx is ar2. The fluent of 3xx is x3, The 2 The fluent of r*x + 3y3ÿ is 3x2 + {y}• The fluent of xa ̄ ̄ ̄1x is 1x". The fluent of my"-'y is 37. When the Root under a Vinculum is a Compound Quantity; and the Index of the part or factor Without the Vinculum, increased by 1, is some Multiple of that Under the Vinculum: Put a single variable letter for the compound root; and substitute its powers and fluxion instead of those of the same value, in the given quantity; so will it be reduced to a simpler form, to which the preceding rule can then be applied. 2 Thus, if the given fluxion be ǹ = (a2 + x2)3⁄4×3x, where 3, the index of the quantity without the vinculum, increased by 1, making 4, which is just the double of 2, the exponent of 2 within the vinculum: therefore, putting z = a2 + x2, thence 2% a2, the fluxion of which is 2xx = ż; hence then r32 = 1ż (z — a2), and the given fluxion F, or (a2 +- x2)3x3x, is = z3ż (z — a2) or = Iz - Jaz3ż; and 2 hence the fluent F is = 5 Or, by substituting the value of z instead of it, the same fluent is 3(a2+x2) × (15.22—3a2), or 3 (a2+x2) × (x2 — ža3). In like manner for the following examples. To find the fluent of ✔a + cx × x3Â¿ To find the fluent of (a + cx)3x2x. To find the fluent of (a + cx2) × dx31⁄2. 38. When there are several Terms, involving Two or more Variable Quantities, having the Fluxion of each Multiplied by the other Quantity or Quantities: Take the fluent of each term, as if there were only one variable quantity in it, namely, that whose fluxion is contained in it, supposing all the others to be constant in that term; then, if the fluents of all the terms, so found, be the very same quantity in all of them, that quantity will be the fluent of the whole. Which is the reverse of the 5th rule for finding fluxions: Thus, if the given fluxion be xy + xỷ, then the fluent of xy is xy, supposing y constant: and the fluent of ry is also ry, supposing r constant: therefore xy is the required fluent of the given fluxion xy + xÿ. |