ed sine x, or the half or trilineal segment contained by an arc with its right sine and versed sine, the diameter being d.

Ex. 1. Putting then the said semiseg. or flu. of (dx-x2) = A, to find the fluent of xi (dx - x2). Here assuming y = (dx - x2), and taking the fluxions, they are, y = (di-2x) (dx - x2); hence zi (dx - 2) = di(dx-x2) - = day; theref. the required flu. fxi (dx-x2), is da-y-da-(dx-x)=B suppose.

3

Ex. 2. To find the fluent of ri (dx-x2), having that of xi(dx-x2) given = B. Here assuming y=x (dx - x2), then taking the fluxions, and reducing, there results ý = (dri-4xi) (dx - x2); hence ri(dx-r2) =

dxi(dx - x2) - y = ds-1y, the flu. theref. of x2i (dx x2) is dB - 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 xn-1(dx-x2) being given = c, then the next, or the flu. of ri(dx-x2)

2n+1

will be

dc

n+2

1
n+2

x-(dx-x2).

84. To find the fluent of such expressions as

a case not included in the table of forms.

√(x2±2ax)'

Put the proposed radical (r2±2ax) = x, or x2 + 2ax = 2; then, completing the square, x2+2ax + a2 = r2 + a2, and the root is x + a = √(x2 + a2). The fluxion of this is

=

zż

√(z2+a2)

; theref.

ż

√(x2+2ax)=√(22+a); the flu.

ent of which, by the 12th form, is the hyp. log. of z + √' (z+a2) = hyp. log. of x + a + √(x2 + 2ax), the fluent

required.

i

given, by the above example, the fluent of

suppose. Assume (x2 + 2ar) = y; then its fluxion is

= y - ai; the fluent of which is y - aa = √(x2+2ax) GA, the fluent sought.

1

xi

Ex. 3. Thus also, this fluent of

√(x2+2ax)

i

being given,

will be found,

the flu. of the next in the series, or √(x2+2ax) by assuming x√(x2 + 2ax) = y; and so on for any other of the same form. As, if the fluent of = c; then, by assuming - √(x2 + x-(x2 + 2ах) —

of

√(x2+2ax)

1

n

2n-1ல் be given √(x2+2ax) 2ax) = y, the fluent 2n-1

Ex. 4. In like manner, the fluent of

given, as in the first example, that of

√(x2-2ax)

xi

√(x2-2ax)

being

may be

found; and thus the series may be continued exactly as in the 3d ex. only taking - 2ax for + 2ax.

85. Again, having given the fluent of

√(2ax-x2)'

1

which,

by p. 326, is X circular arc to radius a and versed

√(2a-22), may be assigned by the same method of continu

ation. Thus,

xi

Ex. 1. For the fluent of √(2ax-x)' , assume √(2ax-x2) = y; the required fluent will be found = - √(2ax-x2)+ or arc to radius a and vers. x.

where a denotes the arc mentioned in the last example.

Ex. 3. And in general the fluent of

ac xn-√(2ax-r2), where c is the fluent of

n

, the next preceding term in the series.

46

86. Thus also, the fluent of i√(x - a) being given, =

(x a), by the 2d form, the fluents of ri√(x – a), ii (x-a), &c. ini√(x-a), may be found. And in general, if the fluent of ri (x-a) = c be given; then

..

87. Also, given the fluent of (x - a)mi, which is

1

m+1x
- (x - a) +1 by the 2d form, the fluents of the series

(x-a)mxi, (x - a)mx2i, &c. ..(x-a)mri can be found.
And in general, the fluent of (x - a)mx-'i being given
= c; then by assuming (x - a)m+x" = y, the fluent
x(x-a)m+пає

of (x - a)mx" is found =

m+n+1

Also, by the same way of continuation, the fluents of ri√(ax) and of zi (a+x)" 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 r, connected with half the coefficient of the middle term with its sign. Thus, put z=x+c: then z2=x2+cx+c2; theref. z? +c2 = x2 + cx, and z2+b-102 = x2 + cx+b the given trinomial; which is = x2 + a2, by putting a2 =

Here z = 1+2; then z2 = x2 + 4x + 4, and z2 + 1 = 5 + 4x + x2, also i = ż; theref. the proposed fluxion re

duces to

3ż

✓(1+22); the fluent of which, by the 12th form

is 3 hyp. log. of z + (1+2) = 3 hyp. log. x + 2 + √(5 + 4x + x2).

Ex. 2. To find the fluent of i (b+cx + dr2) =

id X

C

Here assuming += 2; then i =ż, and the proposed

flux. reduces

2d

b

C2

putting a2 for -4 and the fluent will be found by a

Similar process to that employed in ex. 1, art. 80.

Ex. 3. In like manner, for the flu. of -- √(b+cx" +

1

b Ca d4a

; hence the given

fluxion becomes = 2√ × √ (22 + a2), and its fluent as in

n

the last example.

Ex. 4. Also, for the fluent off

1

X

dn

b+cx+dra; assume

= 2, then the fluxion may be reduced to the form

Fa, 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 x, 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 minimum, or the least, when z 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 remainder or difference, a2 - bx, will increase, as the term ba, 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 a 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 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 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 M. 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 ordinates have several maxima, as st, wx, and several minima, as vu, vz, &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.