of Ex. 2. To find the fluent of B. ✓ia=. Here y must be assumed = x2 √(x+a) ; Jixta, for then taking the flu. and reducing, there is found ya; theref.ƒ 4B x2x x2x v(x+a) - aв = 3x2 √ (x + a) — za (x — 2a) × } √ (x+a) = (9x-4ax+ 8a2) X is √(x+a). In the same manner the student will find the fluents of x2x √(a−x) being given = c, he will find the fluent of And in general, the fluent of xnx 2 78. In a similar way we might proceed to find the fluents of other classes of fluxions by means of other fluents in the table of forms; as, for instance, such as xx√(dx-x2), x2x√(dx-x2), x3 xv (dx-x2), &c. depending on the fluent of xy(dx-x2), the fluent of which, by the 16th tabular form, is the circular semisegment to diame ter d and versed sine x, or the half or trilineal segment con tained by an arc with its right sine and versed sine, the diaméter being d. Ex. 1. Putting then said semiseg. or fluent of x√(dx − x2) =A, to find the fluent of xx(dx-x2. Here assuming y = (dx-x2), and taking the fluxions, they are y = 3(dx-2 xx) √ (dx-x2); hence xx √ (dx-x2) = {dx√ (dx − x2) ~ }y = dy; therefore the required fluent, frx (dx-x2), is da—ży — ¿dâ—†(dx − x2)*——в suppose. Er. 2. To find the fluent of xx(dx-x2), having that of xx √ (dx-x2) given B. Here assuming y=x(dx-x2), then taking the fluxions, and reducing, there results = ({dxx-4x3x) (dx-2); hence xx (dx-x2) = {dxx √(dx—x3) — 4y=dy, the flu. theref. of x2(dx--x2) is fdB—¡y = {dB − {x{dx — x2)}. Ex. 3. In the same manner the series may be continued to any extent; so that in general, the flu. of x/(dx—x2) being given c, then the next, or the flu. of xx(dx—x2) 79. To find the fluent of such expressions as a case not included in the table of forms> Put the proposed radical ✔ (x2 ± 2ax) = z, or x2 +2ax - z2; then, completing the square, x2+2ax+a2= and the root is x±a = √ (z2 + a2). The fluxion of this is x= 2z = ; theref. ✓ (22+a2) ; the ✔ (x2 + 2ax) √(s2+ a2) fluent of which, by the 12th form, is the hyp. log, of z + √ (z2 + a2)= hyp. log. of x ±a + √(x2± 2ax), the fluent required. Ex. 2. To find now the fluent of xx √(x2+2ax)' having given, by the above example, the fluent of suppose. xx tax √(x2+2ax) 11 Assume (x2+2ax)= y; then its fluxion is √(x2 +2ax)=y; theref. a; the fluent of which is y → as = √(x2 + 2ax) —as, by assuming ^/ (x2+2ax) =y; and so on for any other of the same form. As, if the fluent of given, as in the first example, that of √(x2— Zax) being may be found; and thus the series may be continued exactly as in the 3d ex. only taking - 2ax for +2ax. 30. Again, having given the fluent of (ax-x)' is a of which, x circular arc to radius a and versed sine x, the fluents xx ✔(2ax-x2)√(2ax-x2)' by the same method of continuation. Thus, Ex. 1. For the fluent of Ex. 2. In like manner the fluent of x2x is f or arc to radius a and vers. x. is where a denotes the arc mentioned in the last example. Ex. 3. And in general the fluent of 21-1 12 1 ✔ (2ax x2) ac———xn−1 √√ (2ax — x2), where c is the fluent of √(2ax—x2)' 12 - the next preceding term in the series. 81. Thus also, the fluent of (x-a) being given, = (x-a), by the 2d form, the fluents of x x √(x − a), x2x √(x-a), &c. . . x" x √(x-a), may be found. And in general, if the fluent of xx√(x — a) = c be given; then by assuming x”(x~-~-a) z =y, the fluent of x^x √ (x-a) is 82. Also, given the fluent of (x a)m which is m + 1 (x-a)mt1 by the 2d form, the fluents of the series (-a)m xx, (x − a) mx3 x. &c... (x − a)" x1x can be found. And in general, the fluent of (x-am- being given = c; then by assuming (-a)mixy, the fluent of (x - a)mxnx is xn(x—a)m11+nac Also, by the same, way of continuation, the fluents of r(ar) and of "x (ex)" may be found.} 83. 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 z, connected with half the coefficient of the middle term with its sign. Thus, put z=x+c: then z2 = x2+cx+fc2; theref. 2-c2 x + cx, and z2 +6 - tc2 x2 + cx+b the given trinomial which is = z2 + a2, by putting a2 = 6-102. Here zx+2; then z2 = x2 + 4x + 4, and z2 + 1 = 5+4x+x2, also ż; theref. the proposed fluxion re the fluent of which, by the 12th form in this vol. is 3 hyp. log. of z + √(1 + z) = 3 hyp. log, ≈ +2 + √(5 +4.x + x2). Ex. 3. To find the fluent of x (b + cx + dx2)=x√d × Here assuming x + = z; then xz,and the proposed 2d b c2 flux. reduces to ï ✔d× √(z2+7)=√dX√(z2+a2), b c2 putting a2 for d 4d2 ; and the fluent will be found by a sim ilar process to that employed in ex. 1 art, 75. Ex. 3. In like manner, for the flu. of x1x √ (b + cxn + = 1 n dx"), assuming 2+=z, nxnxz, and 1;==; =z, then the fluxion may be reduced to the form 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. 84. 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 a + bx, evidently increases as x, or the term bx, increases; therefore the given expression will be the greatest, or à maximum, when x 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 a-bx, where a and b denote constant or invariable quantities, and a flowing or variable one. Now, it is evident that the value of this remainder 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 - br is a maximum, when x is the least, or nothing at all; and the difference is the least, when x is the greatest. 85. 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 seyeral 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 н. 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 м. And in the curve Noq, the ordinate decreases only to the position op, where it is least, or a minimum; and after that it continually increases towards q But in the curve RSU &c. the prdinates have several maxima, as ST, wx, and several minima, as vu, yz, &c. 51. Now |