86. Thus also, the fluent of (a) being given, = (-a), by the 2d form, the fluents of xi√(x − a), ï2ƒ✓✅ (x − a), &c. . . ï1i ✅ (x — a), may be found. And in general, if the fluent of x-1 √(x − a) = c be given; then 87. Also, given the fluent of (x - a)", which is by the 2d form, the fluents of the series 1 m+1 (x-a)" (x a)mxi, (x a)mx2i, &c. (x-a)"x" can be found. And in general, the fluent of (x-a)" being given = c; then by assuming (x a)m+ny, the fluent a)" is found = __ m+n+1 of (x Also, by the same way of continuation, the fluents of r(ar) and of (a) may be found. 88. When the fluxional expression contains a trinomial quantity, as (b + cx + x2), this may be reduced to a binomial, by substituting another letter for the unknown one x, connected with half the coefficient of the middle term with its sign. Thus, put z=x+c: then 2x2+cx+‡c2; theref. zc2 = x2 + cx, and z2+b2 = x2 + cx+b the given trinomial; which is = 22+ a2, by putting a2 = b - c2. Ex. 1. To find the fluent of 3i Here z=x+2; then z2 = 3 + 4x+4, and z2 + 1 = 5 + 4x + x2, also żż; theref. the proposed fluxion re ; the fluent of which, by the 12th form is 3 hyp. log. of 2 + √ (1 + z) = 3 hyp. log. x + 2 + √(5+ 4x + x2). Here assuming + =z; then i = ż, and the proposed Similar process to that employed in ex. 1, art. 80. Ex. 3. In like manner, for the flu. of x--1 ✔(b + cx2 + 1 =z, nxa—1j = 2, and 2"-12=− ż; 2d 1 4a2 4da)=√d ; hence the given fluxion becomes = ż√d × √(23±a), and its fluent as in the last example. n Ex. 4. Also, for the fluent of 2d xn-li b+cx+dr ; assume =2, then the fluxion may be reduced to the form i X dn x2 +a2' and the fluent found as before. So far on this subject may suffice on the present occasion. But the student who may wish to see more on this branch, may profitably consult Mr. Dealtry's very methodical and ingenious treatise on Fluxions, lately published, from which several of the foregoing cases and examples have been taken or imitated. OF MAXIMA AND MINIMA; OR, THE GREATEST AND LEAST MAGNITUDE OF VARIABLE OR FLOWING QUANTITIES. 89. 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, the expression or sum a2 + bx, evidently increases as r, or the term bx, increases; therefore the given expression will be the greatest, or a maximum, when x is the greatest, or infinite; and the same expression will be a minis mum, 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, it is evident that the value of this re. mainder or difference, a2 — bx, will increase, as the term bx, or as x, decreases; therefore the former will be the greatest, when the latter is the smallest ; that is, a2 bx is a maximum, when x is the least, or nothing at all; and the difference is the least, when x is the greatest. 90. 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 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 NOQ, 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 sт, wx, and several minima, as vu, yz, &c. 91. Now, because the fluxion of a variable quantity, is the rate of its increase or decrease; and because the maximum or minimum, of a quantity neither increases nor decreases, at those points or states; therefore such maximum or minimum has no fluxion, or the fluxion is then equal to nothing. From which we have the following rule. To find the Maximum or Minimum. 92. From the nature of the question or problem, find an algebraical expression for the value, or general state, of the quantity whose maximum or minimum is required; then take the fluxion of that expression, and put it equal to nothing; from which equation, by dividing by, or leaving out, the fluxional letter and other common quantities, and performing other proper reductions, as in common algebra, the value of the unknown quantity will be obtained, determining the point of the maximum or minimum. - So, if it be required to find the maximum state of the compound expression 100x - 5x2±c, or the value of x when 100x 5x2c is a maximum. The fluxion of this expression is 100% 10x; which being made =0, and divided by 10%, the equation is 10 x = 0; and hence x= 10. That is, the value of x is 10, when the expression 100x - 5x2c is the greatest. As is easily tried for if 10 be substituted for x in that expression, it becomes +c+500: but if, for x, there be substituted any other number, whether greater or less than 10, that expression will always be found to be less than c + 500, which is therefore its greatest possible value, or its maximum. 93. It is evident, that if a maximum or minimum be any way compounded with, or operated on, by a given constant. quantity, the result will still be a maximum or minimum. That is, if a maximum or minimum be increased, or decreased, or multiplied, or divided, by a given quantity, or any given power or root of it be taken; the result will still be a maximum or minimum. Thus, if x be a maximum or minimum, then also is x + a or x — a, or ax, or or xa, a or x, still a maximum or minimum. Also, the logarithm of the same will be a maximum or a minimum. And therefore, if any proposed maximum or minimum can be made simpler by performing any of these operations, it is better to do so, before the expression is put into fluxions. 94. When the expression for a maximum or minimum contains several variable letters or quantities; take the fluxion of it as often as there are variable letters; 'supposing first one of them only to flow, and the rest to be constant ; then another only to flow, and the rest constant; and so on for all of them then putting each of these fluxions = 0, there will be as many equations as unknown letters, from which these may be all determined. For the fluxion of the expression must be equal to nothing in each of these cases; otherwise the expression might become greater or less, without alter. ing the values of the other letters, which are considered as constant. So, if it be required to find the values of x and y, when 4x2-xy+ 2y is a minimum. Then we have, First, 8xiiy = 0, and 8x 0, and 2 And hence y or 8x 16. y= 0, or y=8x. =0, or x = 2. x= 95. To find whether a proposed quantity admits of a Maximum or a Minimum. Every algebraic expression does not admit of a maximum or minimum, properly so called; for it may either increase continually to infinity, or decrease continually to nothing; and in both these cases there is neither a proper maximum nor minimum; for the true maximum is that finite value to which an expression increases, and after which it decreases again and the minimum is that finite value to which the expression decreases, and after that it increases again. Therefore, when the expression admits of a maximum, its fluxion is positive before the point, and negative after it but when it admits of a minimum, its fluxion is negative before, and positive after it. Hence then, taking the fluxion of the expression a little before the fluxion is equal to nothing, and again a little after the same; if the former fluxion be positive, and the latter negative, the middle state is a maximum ; but if the former fluxion be negative, and the latter positive, the middle state is a minimum. So, if we would find the quantity ax x2 a maximum or minimum; make its fluxion equal to nothing, that is, ai - 2x = 0, or (a 2x)=0; dividing by, gives a -2x = = 0, or x = a at that state. Now, if in the fluxion (a-2x), the value of x be taken rather less than its true value, a, that fluxion will evidently be positive; but if x be taken somewhat greater than a the value of a - 2x, and consequently of the fluxion, is as evidently negative. Therefore, the fluxion of ax-x2 being positive before, and nega tive after the state when its fluxion is = 0, it follows that at this state the expression is not a minimum, but a maximum. Again, taking the expression x3-ax, its fluxion 32axi(3x-2a) x=0; this divided by xi gives 3x-2a=0, and xa, its true value when the fluxion of x3 — ar2 is equal to nothing. But now to know whether the given expression be a maximum or a minimum at that time, take x a little less than a in the value of the fluxion (3x — 2a) xử, and this will evidently be negative; and again, taking x « |