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conc B is read
mesured on a radius), to the radius of the base: that is, if thes of r be estimated from the centre of the base, and
r be the radius, z will vary as r
3 ~ r = −v (x2 + y2), or r2
the base of a right-angled c
21% + z2 = x22 + y2.
gy, the simplest equation of the conic surface, will be Now from this the nature of curves formed by planes cutting the cone in different directions, may readily be inferred. Let it be supposed, first, that the cutting plane is inclined to through its centre: then wili z = x, and this value of z sub2rr, which is the equation of the projection of the curve sture. for it in the equation of the surface, will give r on the plane of the cone's base: and this (art. 3 of this chap.) Or, taking the thing more generally, let it be supposed that the cutting plane is so situated, that the ratio of r to z shall subsituted for z and z2 in the equation of the surface, will be that of 1 to m: then will mx =
cone in the angle of 45 ̊, and passes
is manifestly an
give, plane equation,
equation to a parabola.
- z, and m2x2 = z2. These
the equation of the projection of the section on the of the base, 72 2mx + (m2 - 1)2= y. Now this if Im be greater than unity, or if the cutting plane between the vertex of the cone and the parabolic secwill be that of an hyperbola: and if, on the contrary, the cutting plane pass between the parabola and the base, i. e. if m be less than unity, the term (m2-1) will be negative, when the equation will obviously designate an ellipse.
Schol. It might here be demonstrated, in a nearly similar manner, that every surface formed by the rotation of any conic section on one of its axes, being cut by any plane whatever, will always give a conic section. For the equation of such surface will not contain any power of x, y, or 2, greater than the second; and therefore the substitution of any values of z in terms of x or of y, will never produce any powers of ror of y exceeding the square. The section therefore must be a line of the second order. See, on this subject, Hutton's Mensuration, part iii, sect. 4.
Ex. 3. Let the equation to the curve surface be xyz=a3.` Then will the curve surface bear the same relation to the solid right angle, which the curve line whose equation is xy=a bears to the plane right angle. That is, the curve surface will be posited between the three rectangular faces bounding such solid right angle, in the same manner as the equilateral hyperbola is posited between its rectangular asymp
And in like manner as there may be 4 equal equila
teral hyperbolas comprehended between the same rectangular asymptotes, when produced both ways from the angular point; so there may be 6 equal hyperboloids posited within the 6 solid right angles which meet at the same summit, and all placed between the same three asymptotic planes.
On the Construction of Equations.
To Construct Simple Equations, Geometrically. HERE the sole art consists in resolving the fractions, to which the unknown quantity is equal, into proportional terms; and then constructing the respective proportions, by means of probs. 8, 9, 10, and 27 Geometry. A few simple examples will render the method obvious.
1. Let x = ; then ca: b: x. Whence may be found by constructing according to prob. 9 Geometry.
2. Let x= First construct the proportion d: a :: b:
3. Let x=2. Then, since a2-b2= (a + b) ×, (a−b) z
it will merely be necessary to construct the proportion c: a + b :: a—b: x.
5. Let x =
i, the difference of those
First find, the fourth proporaf
tional to b, a and ƒ, which make = h. Then x = or, by construction it will be h+c: ad :: a: x.
a2 + 12
. Make the right-angled triangle ABC such
that the leg AB = α, BC = b; then AC = √(AB2 +BC2)=√(a+b2), by th. 34 Geom. Hence x = 10. Construct therefore the proportion
C: AC :: AC: x, and the unknown quantity will be found, as required.
7. Let x =
a2 + cd
mean proportional between AC = c, and CB=d, that is, find CD =√/cd. Then make CE = a, and join DE, which will evidently be = √(a2 + cd). Next on any line EG set off EF = h + C, EG ED; and draw GH parallel to FD, to meet DE (produced if need be) in H. So shall EH ber, the third proportional to h + c, and (a+cd), as required.
Note. Other methods suitable to different cases which may arise are left to the student's invention. And in all constructions the accuracy of the results, will increase with the size of the diagrams; within convenient limits for operation.
To Find the Roots of Quadratic Equations by Construction.
In most of the methods commonly given for the construction of quadratics, it is required to set off the square root of the last term; an operation which can only be performed accurately when that term is a rational square. We shall here describe a method which, at the same time that it is very simple in practice, has the advantage of showing clearly the relations of the roots, and of dividing the third term into two factors, one of which at least may be a whole number. In order to this construction, all quadratics may be classed under 4 forms: viz,
x2 + ax + bc = 0.
ax + bc = 0.
1. One general mode of construction will include the first two of these forms. Let a2 Fax bc = 0, and b be greater than c. Describe any circle ABD having its diameter not less than the given quantities a and b c, and within this circle
inscribe two chords, AB = a, AD = b
c, both from any common assumed point A. Then, produce AD to F so that DF = c, and about the centre c of the former circle, with the radius CF, describe another circle, cutting the chords AD, AB, produced, in F, E, G, H: so shall AG be the affirmative and AH the negative root of the equation x2 + ax-be = 0; and contrariwise AG will be the negative and AH the affirmative root of the equation x2 bc0.
For, AF or AD + DF = b, and DF or AE = c; and, making AG or BH = x, we shall have AH = a + x: and by the property of the circle EGFH (theor. 61 Geom.) the rectangle EA AF GA. AH, or bc (a + x)x, or again by transpo
sition x2+ ax bc
= 0. Also if AH be =-x, we shall have
AG or BH or AH
a: and conseq. GA . AH = So that, whether AG be = x, or
x, we shall always have x2 + ax − bc = 0. And by an exactly similar process it may be proved that AG is the negative, and AH the positive root of r2 -ax bc = 0.. Cor. In quadratics of the form x2 + ax − bc = 0, the positive root is always less than the negative root; and in those of the form r2 0, the positive root is always greater than the negative one.
2. The third and fourth cases also are comprehended under one method of construction, with two concentric circles. Let x2 + ax + bc = 0. Here describe any circle ABD, whose diameter is not less than either of the given quantities a and b + c; and within that circle inscribe two chords AB α, AD = b+c, both from the same point A. Then in AD assume DF = c, and about c the centre of the circle ABD, with the radius CF describe a circle, cutting the chords AD, AB, in the points F, E, G, H: so shall AG, AH, be the two positive roots of the equation rax + bc = 0, and the two negative roots of the equation x2 + ax + bc=0. The demonstration of this also is similar to that of the first
Cor. 1. If the circle whose radius is CF just touches the chord AB, the quadratic, will have two equal roots; which can only happen when 4a2 = bc.
Cor. 2. If that circle neither cut nor touch the chord AB, the roots of the equation will be imaginary; and this will always happen, in these two forms, when be is greater than a2.
To Find the Roots of Cubic and Biquadratic Equations, by Construction.
1. In finding the roots of any equation, containing only one unknown quantity, by construction, the contrivance consists chiefly in bringing a new unknown quantity into that equation; so that various equations may be had, each containing the two unknown quantities; and further, such that any two of them contain together all the known quantities of the proposed equation. Then from among these equations two of the most simple are selected, and their corresponding loci constructed; the intersection of those loci will give the roots sought.
Thus it will be found that cubics may be constructed by two parabolas, or by a circle and a parabola, or by a circle and an equilateral hyperbola, or by a circle and an ellipse, &c and biquadratics by a circle and a parabola, or by a circle and an ellipse, or by a circle and an hyperbola, &c. Now, since a parabola of given parameter may be easily constructed by the rule in cor, 2 th. 4 Parabola, we select the circle and the parabola, for the construction of both biquadratic and cubic equations. The general method applicable to both, will be evident from the following description.
2. Let M'AM'M be a parabola whose axis is AP, M'M'GM a circle whose centre is c and radius CM, cutting the parabola in the points M, M, M", M": from these points draw the ordinates to the axis MP, M'P', M"P", M"P"; and from c let fall cp perpendicularly to the axis; also draw CN parallel to the axis, meeting PM in N. Let AD DCb, Cмn, the parameter of the parabola P, ARX, PM = y. Then (pa. 31) px = y2: also CM2 CNNM2, or n= (x = a)2 + (y b); that is, x2 ± 2ax + a2 + y2 ± 2by + b2 = n2. Substituting in this equation for x, its value and arranging the terms according to the dimensions of y, there will arise
y ± (2pa + p2)y2 ± 2bp2y + (a2 + b2 — n2)p2 = 0, a biquadratic equation, whose roots will be expressed by the ordinates PM, PM', P"M", "M", at the points of intersection of the given parabola and circle.
3. To make this coincide with any proposed biquadratie whose second term is taken away (by cor. theor. 3); assume