« PreviousContinue »
te terminate at the section, the right line that bisects these will bisect all other lines parallel to them; and is therefore called a diameter of the figure, and the bisected lines ordinates, and the intersections of the diameter with the curve vertices; the common intersection of all the diameters the centre; and that diameter which is perpendicular to the ordinates, the vertex. So likewise in higher curves, if two parallel lines be drawn, each to cut the curve in the number of points that indicate the order of the curve; the right line that cuts these parallels so, that the sum of the parts on one side of the line, estimated to the curve, is equal to the sum of the parts on the other side, it will cut in the same manner all other lines parallel to them that meet the curve in the same number of points; in this case also the divided lines are called ordinates, the line so dividing them a diameter, the intersection of the diameter and the curve vertices; the common intersection of two or more diameters the centre; the diameter perpendicular to the ordinates, if there be any such, the axis; and when all the diameters concur in one point, that is the general centre.
Again, the conic hyperbola, being a line of the second order, has two asymptotes; so likewise, that of the third order may have three; that of the fourth, four; and so on: and they can have no more. And as the parts of any right line between the hyperbola and its asymptotes are equal; so likewise in the third order of lines, if any line be drawn cutting the curve and its asymptotes in three points; the sum of two parts of it falling the same way from the asymptotes to the curve, will be equal to the part falling the contrary way from the third asymptote to the curve; and so of higher curves.
Also, in the conic sections which are not parabolic: as the square of the ordinate, or the rectangle of the parts of it on each side of the diameter, is to the rectangle of the parts of the diameter, terminating at the vertices, in a constant ratio, viz, that of the latus rectum, to the transverse diameter. So in non-parabolic curves of the next superior order, the solid under the three ordinates, is to the solid under the three abscissas, or the distances to the three vertices; in a certain given ratio. In which ratio if there be taken three lines proportional to the three diameters, each to each; then each of these three lines may be called a laius rectum, and each of the corresponding diameters a transverse diameter. And, in the common, or Apollonian parabola, which has but one vertex for one diameter, the rectangle of the ordinates is equal to the rectangle of the abscissa and latus rectum : so, in those curves of the second kind, or lines of the third kind which
have only two vertices to the same diameter, the solid under the three ordinates, is equal to the solid under the two abscissas, and a given line, which may be reckoned the latus
Lastly, since in the conic sections where two parallel lines terminating at the curve both ways, are cut by two other parallels likewise terminated by the curve; we have the rectangle of the parts of one of the first, to the rectangle of the parts of one of the second lines, as the rectangle of the parts of the second of the former, to the rectangle of the parts of the second of the latter pair, passing also through the common point of their division. So, when four such lines are drawn in a curve of the second kind, and each meeting it in three points; the solid under the parts of the first line, will be to that under the parts of the third, as the solid under the parts of the second, to that under the parts of the fourth. And the analogy between curves of different orders may be carried much further: but as enough is given for the objects of this work; we shall now present a few of the most useful problems.
Knowing the Characteristic Property, or the Manner of Description of a Curve, to find its Equation.
This in most cases will be a matter of great simplicity; because the manner of description suggests the relation between the ordinates and their corresponding abscissas; and this relation, when expressed algebraically, is no other than the equation to the curve. Examples of this problem have already occurred in sec. 4 ch. i, of this volume: to which the following are now added to exercise the student.
Ex. 1. Find the equation to the cissoid of Diocles; whose manner of description is as below.
From any two points P, S, at equal distances from the extremities A, B, of the diameter of a semicircle, draw, ST, PM, perpendicular to AB. From the point T where ST cuts the semicircle, draw a right line AT, it will cut PM in M, a point of the curve required.
Now, by theor. 87 Geom. AS. SB ST2; and by the construction, As SB AP. PB. Also the similar triangles APM,
AST, give AP PM :: AS: ST :: PB : ST =
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.