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. PROBLEM I. 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 at p. 536, &c. vol. i.: 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 r where sr cuts the semicircle, draw a right line AT, it will cut PM in M, a point of the curve required. M A P S 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 = PM PB Conse AP or PA PB. PM2. Hence, if the diameter AB = PMy; the equation is x3 y' (d - x). = d, AP = 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 M, so that AP: PD: AB: PM: then is м always a point in the curve. d-x Ans. = PROBLEM II. 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 a2x, or y = /a2x, to a line of the third order, be proposed. 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 D parallel to AB. Then, let it be inquired whether the curve passes through the point A, or not. In order to this, we must ascertain what y will be when B M AP C H 0: and in that case_y=3/(a3×0), that is, y=0. There fore 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 x when y = 0; this will be /a2x=0, or x — 0. Consequently the curve does not cut the axis in any other point than A. Make x AP = a; and the given equa. will become y = Va3 a V. There. fore draw PM parallel to AB, and equal to a /, so will м be a point in the curve. Again, make x = AC = a; then the equation will give y=2/a-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 r∞, we shall have y = (ax); that is, y is infinite when x is so; and therefore the curve passes on to infinity. And further, since when z is taken 0, it is also y = 0, and when x ∞, it is also y; 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 rs = AP=a. But PM = a// a/4, which is manifestly greater than a; so that PM is greater than Ps, and consequently the curve is concave to the axis AC. Now, because in the given equation yax the exponent of x is odd, when x 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 x 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 a2x2 — x1. In this equation we have y=±√(a2x2); = where, when x = 0, y B then y√(a—a2)=ja 3; which is the value of the semi-ordinate PM when Ar= 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 (a3 — x2). (x — b)2 = x2y3, or y = ± I ✓ (a2—x2). Y G M M If C PAP Here, if x = 0, then y becomes infinite; 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 A 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 x b = 0, and consequently the curve passes through B, having there a double point. If AP be taken greater than AB, then will there be 12 m 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 - 23 becomes negative, and the value of y impossible so that the curve does not go beyond D. Now let x be considered as negative, or as lying on the side of A towards c. Then y = + +√(a2—x2). Here if x vanish, both these values of y become infinite; and consequently the curve has two indefinite arcs on each side the asymptote or directrix Av. If x increase, y manifestly diminishes; and when xa, 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 x 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 conjugate point. In the figure, m'cm' represents what is termed the superior conchoid, and GBMDMBm 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 b cos of the polar equation z = ±a. It will merely be requisite to set off from any assumed point A, the distance AB = 6; then to draw through в a right line mLM' 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 m' and m be points in the superior and inferior conchoids respectively. Ex. 4. Let the principal properties of the curve whose equation is yxn = a+ 1, be sought; when n is an odd number, and when n is an even number. Ex. 5. Describe the line which is defined by the equation xy+ay + cy= bc + bx. Ex. 6. Let the Cardioide, whose equation is y — Cay3 + (2x2 + 12a2) y2 — (6ax2 + 8a3) y + (x2 + 3a2) x2 = 0, be proposed. . Ex. 7. Let the Trident, whose equation is xy = ax3 + bx + cx+d, be proposed. Ex. 8. Ascertain whether the Cissoid and the Witch, whose equations are found in the preceeding problem, have asymptotes. PROBLEM III. 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, гм, мn, 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 F G M P |