62. The second fluxion of xy, when is constant, is 63. The second fluxion of an is 64. The third fluxion of r", 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 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 x = y, then is j; or if x : y :: n: I, then is * : j :: n : 1; or if x = ny, then is xnj. 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 ay + 2y 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 divide by the index 6, and it is 3x5x. 3x5; which is the fluent of the proposed fluxion 3xx. In like manner, The fluent af 2axx is ax2. The fluent of 3x2x is x3. U 2 The I 2 2 The fluent of xx+3y3ÿ is 3.x2 + 93. The fluent of axis. 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 F = (a2 + x2)3⁄43μ3x, 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 za2+x', thence-x2 2 a2, the fluxion of which is 2xx =ż; hence then 3 = r2 = (≈ — a2), and the given fluxion F, or - Or, by substituting the value of z instead of it, the same fluent is 3(a2 + x2)3× (15x2−5·5a2), or ‚¿‚a2+x®}3 „‚x2 — ža2. In In like manner for the following examples. To find the fluent of /a+ cx × x?x. To find the fluent of (a + cx2)3 × dx3x. CZ To find the fluent of ___czż___ or (a + z) ̃3czż. To find the fluent of *\/a2 + z2 or (a2 + z2)2 ̄z ̄ˆ¿. 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 ferm; 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 + xj, then the fluent of xy is xy, supposing y constant: and the fluent of xy is also xy, supposingr constant: therefore xy is the required fluent of the given fluxion xy + xy. கர் 39. When the given Fluxional Expression is in this Form * — ** namely, a Fraction, including Two Quantities, being the Fluxion of the former of them drawn into the latter, minus the Fluxion of the latter drawn into the former, and divided by the Square of the latter: Then, the fluent is the fraction, or the former quantity divided by the latter. That is, xy xy y The fluent of żyj is. And, in like manner, y The fluent of 2.xxy - 2x2y is Though, indeed, the examples of this case may be performed by the foregoing one. Thus, the given fluxion reduces to xy xy xjy; of which, y -, supposing y constant; and y therefore, by that case, is the fluent of the whole * is the fluent of the whole *y-x 40. When the Fluxion of a Quantity is Divided by the Quantity itself: Then the fluent is equal to the hyperbolic logarithm of that quantity; or, which is the same thing, the fluent is equal to 2.30258509 multiplied by the common logarithm of the same quantity. is 2 x hyp. log. of x, or hyp. log. x2. is a × hyp. log x, or = hyp. log. of x2. a + x a + 23 3 is is 41. Many |