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or PA3 PB. PM2. Hence, if the diameter AB = d, AP=x, PMy; the equation is 23y2(d-x).
The complete cissoid will have another branch equal and similar to AMQ, but turned contrary ways; being drawn by means of points T' falling in the other half of the circle. But the same equation will comprehend both branches of the curve; because the square of -y, as well as that of +y, is positive.
Cor. All cissoids are similar figures; because the abscissæ and ordinates of several cissoids will be in the same ratio, when either of them is in a given ratio to the diameter of its generating circle.
Ex. 2. Find the equation to the logarithmic curve, whose fundamental property is, that when the abscissas increase or decrease in arithmetical progression, the corresponding ordinates increase or decrease in geometrical progression.
Ans. y=a*, a being the number whose logarithm is 1, in the system of logarithms represented by the curve.
Ex. 3. Find the equation to the curve called the Witch, whose construction is this: a semicircle whose diameter is AB being given; draw, from any point P in the diameter, a perpendicular ordinate, cutting the semicircle in D, and terminating in м, so that AP: PD :: AB: PM; then is M always a point in the curve.
Ans. y = dv
Given the Equation to a Curve, to Describe it, and trace its Chief Properties.
The method of effecting this is obvious: for any abscissas being assumed, the corresponding values of the ordinates become known from the equation; and thus the curve may be traced, and its limits and properties developed.
Ex. 1. Let the equation y3 of the third order, be proposed.
: a2x', or y = Va2x, to a line
First, drawing the two indefinite lines BH, DC, to make an angle BAC equal to the assumed angle of the co-ordinates; let the values of x be taken upon AC, and those of y upon AB, or upon lines parallel to AB. Then, let it be enquired whether the curve passes through the point A, or not. In order to this, we must ascertain what y will be when
x=0: and in that case y = 3(a2 × 0), that is, y=0. Therefore the curve passes through A. Let it next be ascertained whether the curve cuts the axis ac in any other point; in order to which, find the value of r when y = 0: this will be Ya'x = 0, or x=0. Consequently the curve does not cut the axis in any other point than A. Make r AP, and the given equa. will become y = /a3 = a. Therefore draw PM parallel to AB and equal to a 3/4, so will м be a point in the curve. Again, make x = AC = a; then the equation will give y3⁄43 = a. Hence, drawing CN parallel to AB, and equal to AC or a, N will be another point in the curve. And by assuming other values of y, other ordinates, and consequently other points of the curve, may be obtained. Once more, making x infinite, or x=∞, we shall have y= a2x cc); that is, y is infinite when x is so; and therefore the curve passes on to infinity. And further, since when r is taken = O, it is also y = 0, and when r∞, it is also yoo; the curve will have no asymptotes that are parallel to the co-ordinates.
Let the right line AN be drawn to cut PM (produced if necessary) in s. Then because CN=AC, it will be PSAP1⁄4α. But PM a 3/4, which is manifestly greater than 4a; so that PM is greater than PS, and consequently the curve is concave to the axis AC.
Now, because in the given equation y3 = a'r the exponent of r is odd, when r is taken negatively or on the other side of A, its sign should be changed, and the reduced equation will then be y-ax. Here it is evident that, when the values of a are taken in the negative way from A towards D, but equal to those already taken the positive way, there will result as many negative values of y, to fall below AD, and each equal to the corresponding values of y, taken above AC. Hence it follows that the branch AM'N' will be similar and equal to the branch AMN; but contrarily posited.
Ex. 2. Let the lemniscate be proposed, which is a line of the fourth order, denoted by the equation ay
In this equation we have y=(a— ‚x2) ;. where, when x = 0, y = 0, therefore the curve passes through A, the point from which the values of x are measured. When r±a, then y=0; therefore the curve passes through B and c, supposing AB and AC each a. If x were assumed greater than a, the value of y would become imaginary; therefore no part of the curve lies beyond в or c. When x =
a2x2 — xa.
then y = Va-aa3; which is the value of the semi-ordinate PM when AP AB. And thus, by assuming other values of x, other values of y may be ascertained, and the curve described. It has obviously two equal and similar parts, and a double point at A. A right line may cut this curve in either 2 points, or in 4: even the right line BAC is conceived to cut it in 4 points; because the double point a is that in which two branches of the curve, viz, map, and naq, are intersected.
Ex. 3. Let there be proposed the Conchoid of the ancients, which is a line of the fourth order defined by the equation (a2 — x2)'. (x — b)2 = x2y2, or y = ±2= √(a2 — x13).
Here, if x = 0, then y becomes in
finite; and therefore the ordinate at
A (the origin of the abscissas) is an asymptote to the curve.
If AB = = b, and P be taken between and B, then shall PM and pm be equal, and lie on different sides of the abscissa AP. x=b, then the two values of y vanish, because xb0, and consequently the curve passes through B, having there a double point. If AP be taken greater than AB, then will there be
two values of y, as before, having contrary signs; that value which was positive before being now negative, and vice versa. But if AD be taken = a, and P comes to D, then the two values of y vanish, because in that case / (a2 — x2) = 0. If ap be taken greater than AD or a, then a2-2 becomes negative, and the value of y impossible: so that the curve does not go beyond D.
Now let r be considered as negative, or as lying on the side of A towards c. Then y = √(a2x2). Here
if a vanish, both these values of y become infinite; and consequently the curve has two indefinite arcs on each side the asymptote or directrix AY. If increase, y manifestly diminishes; and when ra, then y vanishes: that is, if Ac=AD, then one branch of the curve passes through c, while the other passes through D. Here also, if a be taken greater than a, y becomes imaginary; so that no part of the curve can be found beyond c.
If a = b, the curve will have a cusp in B, the node between B and D vanishing in that case. If a be less than b, then B will become a conjugste point.
In the figure, m'cm' represents what is termed the superior conchoid, and GBMDMвm the inferior conchoid. The point B is called the pole of the conchoid; and the curve may be readily constructed by radial lines from this point, by means of the polar equation z = a. It will merely be requisite to set off from any assumed point A, the distance AB b; then to draw through в a right line mLм' making any angle with CB, and from L, the point, where this line cuts the directrix AY (drawn perpendicular to CB) set off upon it LM = Lm = a; so shall м' and m be points in the superior and inferior conchoids respectively.
Ex. 4. Let the principal properties of the curve whose equation is ya"a"+1, be sought; when n is an odd number, and when n is an even number.
Er. 5. Describe the line which is defined by the equation xy+ay + cy = bc + bx.
Ex. 6. Let the Cardioide, whose equation is y+- 6ay3 + (2.x2 + 12a2) y3 — (6ax2 + 8a3)y + (x2 + 3u2)x12 = 0, be proposed.
Ex. 7. Let the Trident, whose equation is xy = ax3 + bx2 + cx+d, be proposed.
Ascertain whether the Cissoid and the Witch, whose equations are found in the preceding problem, have asymptotes.
To determine the Equation to any proposed Curve Surface. Here the required equation must be deduced from the law or manner of construction of the proposed surface, the reference being to three co-ordinates, commonly rectangular ones, the variable quantities being x, y, and z. Of these, two, namely x and y, will be found in one plane, and the third z will always mark the distance from that plane.
Ex. 1. Let the proposed surface be that of a sphere, FNG. 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 r. PM =Y, MN = %; the right-angled triangle APM will give
Then, if AP = x,
AM2 = AP2 + PM2 = x2 + y2. In like manner, the rightangled triangle AMN, posited in a plane perpendicular to the former, will give ANAM + MN, that is, 2x2+y2+22; or z2 = p2 — x2 — y', the equation to the spherical surface, as required.
Scholium. Curve surfaces, as well as plane curves, are arranged in orders according to the dimensions of the equations, by which they are represented. And, in order to determine the properties of curve surfaces, processes must be employed, similar to those adopted when investigating the properties of plane curves. Thus, in like manner as in the theory of curve lines, the supposition that the ordinate y is equal to O, gives the point or points where the curve cuts its axis; so, with regard to curve surfaces, the supposition of z=0, will give the equation of the curve made by the intersection of the surface and its base, or the plane of the coordinates x, y. Hence, in the equation to the spherical surface, when ≈ = 0, we have x2 + y2 = r2, which is that of a circle whose radius is equal to that of the sphere. See p. 31. Ex. 2. Let the curve surface proposed be that produced by a parabola turning about its axis.
Here the abscissas x being reckoned from the vertex or summit of the axis, and on a plane passing through that axis; the two other co-ordinates being, as before, y and z; and the parameter of the generating parabola being p: the equation of the parabolic surface will be found to be z2 + y2 px = 0.
Now, in this equation, if z be supposed = 0, we shall have ypr, which (pa. 31) is the equation to the generating parabola, as it ought to be. If we wished to know what would be the curve resulting from a section parallel to that which coincides with the axis, and at the distance a from it, we must put z = a; this would give y2 = px a2, which is still an equation to a parabola, but in which the origin of the abscissas is distant from the vertex before assumed by the quantity p'
Ex. 3. Suppose the curve surface of a right cone were proposed.
Here we may most conveniently refer the equation of the surface to the plane of the circular base of the cone. In this case, the perpendicular distance of any point in the surface from the base, will be to the axis of the cone, as the distance of the foot of that perpendicular from the circumference (measured