S 0. Assume also p

ya — qy3 ‡ry — 8 —
1; then comparing
the terms of the two equations, it will be, 2a-1=q, or a===

9+1

2

2.

2b=r, or b
; a2 + b2 — n2
s, or n2=a2+
bas, and consequently n=(a2+b2+s). Therefore de-
scribe a parabola whose parameter is 1, and in the axis take

AD=

q+1; at right angles to it draw Dc and r; from

the centre c, with the radius (a2 + b2 + s), describe the
circle м'n'GM, cutting the parabola in the points M, M, и", M";
then the ordinates PM, P'M', P"M", "M", will be the roots re-
quired.

Note. This method, of making p1, has the obvious ad-
vantage of requiring only one parabola for any number of
biquadratics, the necessary variation being made in the radius
of the circle.

Cor. 1. When Dc represents a negative quantity, the ordinates on the same side of the axis with e represent the negative roots of the equation; and the contrary.

Cor. 2. If the circle touch the parabola, two roots of the equation are equal; if it cut it only in two points, or touch it in one, two roots are impossible; and if the circle fall wholly within the parabola, all the roots are impossible.

;

Cor. 3. If a2 + b2 = n2, or the circle pass through the
point A, the last term of the equation, i. e. a2+b2 — n2 )p2=0
and therefore y2+(2pa+p2)y2+2bp3y=0, or
y3±(2pa + p3)y2bp2=0. This cubic equation may be
made to coincide with any proposed cubic, wanting its second
term, and the ordinates PM, P'M", P"M", are its roots.
Thus, if the cubic be expressed generally by y3±qy±s=0.
By comparing the terms of this and the preceding equation,
we shall have ± 2pa + p2 +9, and ± 2bp2

2p

S

2p2

3

±s, or So that, to construct a cubic

equation, with any given parabola, whose half parameter is
AB (see the preceding figure); from the point B take in the
axis, (forward if the equation have -g, but backward if q be
positive) the line BD= ; then raise the perpendicular DC

2p2

9 2p

and from c'describe a circle passing through the ver

tex A of the parabola; the ordinates PM, &c. drawn from the
points of intersection of the circle and parabola, will be the
roots required.

PROBLEM

PROBLEM IV.

To Construct an Equation of any Order by means of a Locus
of. the same Degree as the Equation proposed, and a Right

Line.

3

As the general method is
the same in all equations, let
it be one of the 5th degree, as
x2 - bx1 +αcx3 - a2 dx2+a3
ex-a4f=0. Let the last term
a4f be transposed; and, tak-
ing one of the linear divisors,
f, of the last term, make it
equal to z, for example, and divide the equation by a4; then
x5 · bx2 +αcx3 — a2 dx2+a3 ex

will z=

3

4 α

On the indefinite line BQ describe the curve of this equation, BMDRLFC, by the method taught in prob. 2, sect. 1, of this chapter, taking the values of x from the fixed point B. The ordinates PM, SR, &c. will be equal to z; and therefore, from the point в draw the right line BA =ƒ, parallel to the ordinates PM, SR, and through the point A draw the indefinite right line Kс both ways, and parallel to BQ. From the points in which it cuts the curve, let fall the perpendiculars, MP, RS, CQ: they will determine the abscissas BP, BS, EQ, which are the roots of the equation proposed. Those from A towards a are positive, and those lying the contrary way are negative.

If the right line ac touch the curve in any point, the cor responding abscissa x, will denote two equal roots; and if it do not meet the curve at all, all the roots will be imaginary. If the sign of the last term, a4f, had been positive, then we must have made zf, and therefore must have taken BA=-f, that is, below the point P, or on the negative side.

EXERCISES.

Ex. 1. Let it be proposed to divide a given arc of a circle into three equal parts.

Suppose the radius of the circle to be represented by r, the sine of the given arc by a, the unknown sine of its third part by x, and let the known arc, be 3u, and of course, the required arc be u. Then, by equa. VIII, IX, chap. iii, we shall have

!

Putting, in the first of these equations, for sin 3u its given value a, and for sin 2u, cos 2u, their values given in the two other equations, there will arise

Then substituting for sin u its value x, and for cos2 u its value r2 - x2 and arranging all the terms according to the powers of x, we shall have

3

2

3

a cubic equation of the form x3-px+q=0, with the condition
that p3 193; that is to say, it is a cubic equation falling
under the irreducible case, and its three roots are represent-
ed by the sines of the three arcs u, u + 120o, and u+240o.
Now, this cubic may evidently be constructed by the rule
in prob. 3 cor 3. But the trisection of an arc may also be
effected by means of an equilateral hyperbola, in the follow-
ing manner.

Let the arc to be trisected be ab.
In the circle ABC draw the semi-
diameter AD, and to AD as a diame- H
ter, and to the vertex A, draw the
equilateral hyperbola AE to which
the right line AB (the chord of the

arc to be trisected) shall be a tangent in the point a; then the arc AF, included within this hyperbola, is one third of the

arc AB.

For, draw the chord of the arc AE, bisect AD at a, so that G will be the centre of the hyperbola, join DF, and draw GH parallel to it, cutting the chords AB, AF, in I and K. Then, the hyperbola being equilateral, or having its transverse and conjugate equal to one another, it follows from Def. 16 Conic Sections, that every diameter is equal to its parameter, and from cor. theor, 2 Hyperbola, that GK KIAK2, or that GK : AK :: AK : KI; therefore the triangles GKA, AKI are similar, and the angle KAI AGK, which is manifestly Now the angle ADF at the centre of the circle being equal to KAI or FAB; and the former angle at the centre being measured by the arc AF, while the latter at the circumference is measured by half гB; it follows that AF=1FB, or = ¦ AB, as it ought to be.

=

ADF.

Ex. 2. Given the side of a cube, to find the side of another of double capacity.

Let the side of the given cube be a, and that of a double one y, then 2a3y3, or by putting 2a=b, it will be a2=by3; there are therefore to be found two mean proportionals be

tween

tween the side of the cube and twice that side, and the first
of those mean proportionals will be the side of the double
cube. Now these may be readily found by means of two pa-
rabolas; thus :

Let the right lines AR, AS, be joined
at right angles; and a parabola Aмн be
described about the axis AR, with the
parameter a; and another parabola AMI
about the axis As, wth the parameter b:
cutting the former in м.
Then APC,
PM⇒y, are the two mean proportionals
of which y is the side of the double cube required.

For, in the parabola AMH the equation is y2=ax, and in the parabola AMI it is x2by. Consequently ay yx, and y : x : : x : b. Whence yrab; or, by substitution, y by ab, or by squaring y3b-a2b2; or lastly, y3a2 b— 2a3, as it ought to be.

3

1

THE DOCTRINE OF FLUXIONS.

DEFINITIONS AND PRINCIPLES.

Art. 1. In the Doctrine of Fluxions, magnitudes or quantities of all kinds are considered, not as made up of a number of small parts, but as generated by continued motion, by means of which they increase or decrease. As, a line by the motion of a point; a surface by the motion of a line; and a solid by the motion of a surface. So likewise, time may be considered as represented by a line, increasing uniformly by the motion of a point. And quantities of all kinds whatever, which are capable of increase and decrease, may in like manner be represented by geometrical magnitudes, conceived to be generated by motion.

2. Any quantity thus generated, and variable, is called a Fluent, or a Flowing Quantity. And the rate or proportion according to which any flowing quantity increases, at any position or instant, is the Fluxion of the said quantity at that position or instant and it is proportional to the magnitude by which the flowing quantity would be uniformly increased in a given time, with the generating celerity uniformly continued during that time.

3. The small quantities that are actually generated, produced, or described, in any small given time, and by any continued motion either uniform or variable, are called Increments.

4. Hence, if the motion of increase be uniform, by which increments are generated, the increments will in that case be proportional, or equal, to the measures of the fluxions: but if the motion of increase be accelerated, the increment so generated, in a given finite time, will exceed the fluxion: and if it be a decreasing motion, the increment, so generated, will be less than the fluxion. But if the time be indefinitely small, so that the motion be considered as uniform for that instant; then these nascent increments will always be proportional, or equal, to the fluxions, and may be substituted instead of them, in any calculation.