« PreviousContinue »
In the figure, M'cm' represents what is termed the sup conchoid, and GBMDMBm the inferior conchoid. The B is called the pole of the conchoid; and the curve n readily constructed by radial lines from this point, by of the polar equation z = I a. It will merely quisite to set off from any assumed point A, the AB b; then to draw through в a right line mLM any angle with CB, and from L, the point, where cuts the directrix Ay (drawn perpendicular to CB) set it LM = Lm = a; so shall м' and in be points in the and inferior conchoids respectively.
Ex. 4. Let the principal properties of the cu equation is ya" = "+', be sought; when n is an ber, and when n is an even number.
Ex. 5. Describe the line which is defined by t xy + ay + cy = bc + bx.
Ex. 6. Let the Cardioide, whose equation is (2.x2 + 12a2) y2 (6ax2+8a3)y + (x2 + 3α2 proposed.
Ex. 7. Let the Trident, whose equation i bx2 + cx +d, be proposed.
Ex. 8. Ascertain whether the Cissoid an whose equations are found in the preceding asymptotes.
To determine the Equation to any propose
Here the required equation must be dedu or manner of construction of the proposed s ence being to three co-ordinates, commonly the variable quantities being x, y, and z. namely x and y, will be found in one plan will always mark the distance from that p
Ex. 1. Let the proposed surface be tha The position of the fixed point A, which is the origin of the co-ordinates AP, PM, MN, being arbitrary; let it be supposed, for the greater convenience, that it is at the centre of the sphere. Let MA, NA, be drawn, of which the latter is manifestly equal to the radius of the sphere, and may be denoted by y, MN = x; the right-angled
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.
al 1. Let x = ; then c:a :: 6 : x. Whence x may be found by constructing according to prob. 9 Geometry. 2. Let r = First construct the proportion d : a : which 4th term call g; then x = ; or e:ci:g :x,
a? - 2 3. Let x = a79. Then, since a* – b*= (a +b)x.(a−b);
baa,b it will merely be necessary to construct the proportion cia+b::a-6: x.
a?), bc? 4. Let x =
Find, as in the first case, g =
Then find by the first case i = So shall x = g - i, the difference of those lines, found by construction. 5. Let x = First find , the fourth propor
a(4-d) tional to b, a and f, which make :h. Then I =
; or, by construction it will be h +c:a-d :: a : x.
Make the right-angled triangle ABC such
(measured on a radius), to the radius of the base: that i the values of r be estimated from the centre of the base, r be the radius, z will vary as r√(2+2). Cr quently, the simplest equation of the conic surface, wi 2 ~ r = = √(x2 + y2), or r2 21% + z2 x2 + y2. Now from this the nature of curves formed by plane ting the cone in different directions, may readily be in Let it be supposed, first, that the cutting plane is incl the base of a right-angled cone in the angle of 45°, through its centre: then will zr, and this value o stituted for it in the equation of the surface, will giv 2rry, which is the equation of the projection of t! on the plane of the cone's base: and this (art. 3 of th is manifestly an equation to a parabola.
Or, taking the thing more generally, let it be supp the cutting plane is so situated, that the ratio of be that of 1 to m: then will mx= z, and m2x2 substituted for z and z2 in the equation of the st give, for the equation of the projection of the sec plane of the base, 2mx+ (m2 = 1),x2 = y. 712 equation, if m be greater than unity, or if the c pass between the vertex of the cone and the pa tion, will be that of an hyperbola: and if, on t the cutting plane pass between the parabola and if m be less than unity, the term (m2— 1)x2 wil when the equation will obviously designate an
Schol. It might here be demonstrated, in a lar manner, that every surface formed by the r conic section on one of its axes, being cut by an ever, will always give a conic section. For t such surface will not contain any power of x, than the second; and therefore the substitutio of z in terms of r or of y, will never produce r or of y exceeding the square. The section be a line of the second order. See, on this s Mensuration, part iii, sect. 4.
Er. 3. Let the equation to the curve surf Then will the curve surface bear the sam solid right angle, which the curve line w ry a bears to the plane right angle. surface will be posited between the three bounding such solid right angle, in the s equilateral hyperbola is posited between its totes. And in like manner as there may
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 r? + ar-be =0; and contrariwise AG will be the negative and an the affirmative root of the equation x?
bc = 0. 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 transposition x2 + ax — bc = 0. Also if ah be = -x, we shall have AG or BH or AH
a: and conseq. GA , AH = x2 + ax, as before. So that, whether AG be = x, or AH = - 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 an the positive root of 22 - ar – 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 ra
bc = 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
Fax + 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 = a, AD = b + c, both from the same point A. Then in AD assume di = , and about c the centre of the circle ABD, with the radius of 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 72
ax + bc
- O, 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 ck just touches the chord AB, the quadratic will have two equal roots; which can only happen when ja? = 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 bc is greater than a.
PROBLEM PROBLEM III.
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 con. sists 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 iwo 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 buth, will be evident from the following description.
2. Let m'am'm be a parabola whose axis is AP, M"M'm a circle whose centre is c and radius CM, cutting the pa
İM rabola in the points M, m', m”, m":
JP from these points draw the ordinates to the axis mp, M'P', m"P", "P"; and from c let fall cd perpendicularly to the axis; also draw on parallel to the axis, meeting om in n.
Let AD = 0, DC = b, cm = n, the parameter of the parabola = P, AP = x, PM = y. Then (pa. 31) px = y*: also cm = CN + NM’, or n* = (x 4 a) + (y + b)'; that
? n2 ? is, x2 + 2ax + a + 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
y4 + (2pa + ply* † 2bpy + (Q2 + b2 - n?)p? = 0, a biquadratic equation, whose roots will be expressed by the ordinates PM, P'M', p"m", p"M"", at the points of intersection of the given parabola and circle.
3. To make this coincide with any proposed biquadratic whose second term is taken away (by cor. theor. 3); assume