To Correct the Fluent of any Given Fluxion. 46. The fluxion found from a given fluent, is always perfect and complete; but the fluent found from a given Auxion is not always so; as it often wants a correction, to make it contemporaneous with that required by the problem under consideration, &c: for, the fluent of any given fluxion, as & may be either x, which is found by the rule, or it may be x + c, or x c, that is x plus or minus some constant quantity ; because both x and r±c have the same fluxion *, and the finding of the constant quantity c, to be added or subtracted with the fluent as found by the foregoing rules, is called correcting the fluent. Now this correction is to be determined from the nature of the problem in hand, by which we come to know the relation which the fluent quantities have to each other at some certain point or time. Reduce, therefore, the general fluential equation, supposed to be found by the foregoing rules, to that point or time; then if the equation be true, it is correct; but if not, it wants a correction; and the quantity of the correction, is the difference between the two general sides of the equation when reduced to that particular point. Hence the general rule for the correction is this : Connect the constant, but indeterminate, quantity c, with one side of the fluential equation, as determined by the foregoing rules; then, in this equation, substitute for the variable quantities, such values as they are known to have at any particular state, place, or time; and then, from that particular state of the equation, find the value of c, the constant quantity of the correction. EXAMPLES. 47. ΕΧΑΜ. 1. To find the correct fluent of & = ax3. The general fluent is z = axt, or z = ax + c, taking in the correction c. Now, if it be known that z and x begin together, or that z is = 0, when x = 0; then writing O for both x and z, the general equation becomes 0 = 0 + c, or = c; so that, the value of c being 0, the correct fluents are z = art. But if z be= 0, when x is = b, any known quantity; then substituting 0 for z, and b for x, in the general equation, it becomes 0 = ab + c, and hence we find c = which being written for c in the general fluential equation, it becomes z = ax - ab, for the correct fluents. ab; Or, t Or, if it be known that z is = some quantity d, when x is = some other quantity as b; then substituting d for z, and b for x, in the general fluential equation z = ax + c, it becomes d = ab + c; and hence is deduced the value of the correction, namely, c = d - ab; consequently, writing this value for c in the general equation, it becomes z=ax+ - ab+ + d, for the correct equation of the fluents in this case. 48. And hence arises another easy and general way of correcting the fluents, which is this: In the general equation of the fluents, write the particular values of the quantities which they are known to have at any certain time or position; then subtract the sides of the resulting particular equation from the corresponding sides of the general one, and the remainders will give the correct equation of the fluents sought. So, the general equation being z = ax^; ab+ + d, the correct fluents as before. EXAM. 2. To find the correct fluents of & = 5xx; z being = 0 when x is = a. EXAM. 3. To find the correct fluents of z = 3x √ a + xi z and r being = 0 at the same time. ΕΧΑΜ. 4. Τo find the correct fluent of & = posing z and 3 to begin to flow together, or to be each = 0 at the same time. 2x ΕΧΑΜ. 5. Τo find the correct fluents of & = posing z and a to begin together. OF MAXIMA AND MINIMA; OR, THE GREATEST AND LEAST MAGNITUDE OF VARIABLE OR FLOWING QUANTITIES. 49. MAXIMUM, denotes the greatest state or quantity attainable in any given case, or the greatest value of a variable quantity: by which it stands opposed to Minimum, which is the least possible quantity in any case. Thus, Thus, the expression or sum a2 + br, evidently increases as x, or the term br, increases; therefore the given expression will be the greatest, or a maximum, when a is the greatest, or infinite: and the same expression will be a minimum, or the least, when x is the least, or nothing. Again, in the algebraic expression a2 - bx, where a and b denote constant or invariable quantities, and x a flowing or variable one. Now, it is evident that the value of this remainder or difference, a2 - bx, will increase, as the term br, or as x, decreases; therefore the former will be the greatest, when the latter is the smallest; that is a2 - br is a maximum, when x is the least, or nothing at all; and the difference is the least, when x is the greatest. 50. Some variable quantities increase continually; and so have no maximum, but what is infinite. Others again decrease continually; and so have no minimum, but what is of no magnitude, or nothing. But, on the other hand, some variable quantities increase only to a certain finite magnitude, called their Maximum, or greatest state, and after that they decrease again. While others decrease to a certain finite magnitude, called their Minimum, or least state, and afterwards increase again. And lastly, some quantities have several maxima and minima. Thus, for example, the ordinate BC of the parabola, or such-like curve, flowing along the axis AB from the vertex A, continually increases, and has no limit or maximum. And the ordinate GF of the curve EFH, flowing from E towards H, continually decreases to nothing when it arrives at the point H. But in the circle ILM, the ordinate only increases to a certain magnitude, namely, the radius, when it arrives at the middle as at KL, which is its maximum ; and after that it decreases again to nothing, at the point M. And in the curve now, the ordinate decreases only to the position or, where it is least, or a minimum; and after that it continually increases towards q. But in the curve RSU &c, the ordinates have several maxima, as st, wx, and several minima, as vu, yz, &c. 51. Now, 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 aj + 2j 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 re verse of the Ist rule for finding fluxions. divide by the index 6, and it is or x, which is the fluent of the proposed fluxion 3x5x. In like manner, The fluent of 2arx is ar. The fluent of 3x2x is x3, The 1 √(a-x) 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. Thus, if the given fluxion be F = (a2 + x2x3, 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 r2 within the vinculum: therefore, putting z = a2 + x2, thence x2=x a2, the fluxion of which is 2xx = 2; hence then xx = x2żż (z - a2), and the given fluxion F, or (a2 + v2) 3x3x, is = zz (z-a) or = 12* - aziz; and 2 = 5 5 hence the fluent F is 23-az = 3(2-1). Or, by substituting the value of z instead of it, the same fluent is 3(a2+x2) × (-a), or(a2+x2) x(x2-a2). 1 |