the numerator and denominator of the lowest degree are of two dimensions, therefore we must differentiate these terms twice; hence Hence the origin is a triple point, the axis of x touching one branch, and the tangents of the two other intersecting branches are inclined to the axis of x at angles tan '(+ √2) and tan −√2), as in the annexed figure. In the final differentiation certain terms involving x and y have been rejected, hence it is evident that the multiplicity of a multiple point at the origin may be determined very simply by rejecting certain terms of the equation to the curve and retaining only those of lowest degree. Thus in the present example we may reject the term 1 and retain the others; hence 2 a x2y — a y3 = = 0; which are the equations of the tangents to the curve at the origin; hence, as before, the origin is a triple point. --- Ex. 2. Let the equation of the curve be y=c+ (x − a)2√ (x − b). Freeing the equation from radicals, we get dy = (x − a)/(10x-4a6b); and hence the curve has a double point by osculation, as in the annexed figure. From the equation of the curve we see that when x = a, y = c, and when x = b, y = c, which are the coordinates of the points B and A respectively. III.- Of Cusps and Isolated Points. 74. A cusp, or point of regression, is a double point of the second kind in which the two touching branches terminate, and do not extend beyond this point in one direction. There are two species of cusps, the ceratoid, so called from its resemblance to the horns of animals, the curvature of the two branches lying in opposite directions; and the ramphoid, so called from its resemblance to the beak of a bird, the curvature of the two branches lying in the same direction. The two annexed figures represent the two different kinds of cusps, the former being an instance of a ceratoid and the latter of a ramphoid. The nature of the cusp will be determined by the direction of the curvature of the two branches, as in Art. 64. If the values of d2 y have different signs, the cusp is of the first species; and if they have the same signs, the cusp is of the second species. 75. A conjugate or isolated point is a point the coordinates of which satisfy the equation to the curve, while if to either x or y a value be assigned, differing ever so little from its value at the point, the corresponding value of y or x will be impossible. Ex. I. Let the equation of the curve be (y — x2)2 = x3. 5 Resolving the equation for y, we get y = x2x2, and when x is negative y is imaginary, therefore the curve lies on one side of the origin. Now from the equation of the curve, when x = x = 0, dy dx = 0, and when 0, therefore the axis of x is a common tangent, and the two branches are both convex towards the axis, since when x = O, the dy value of is positive, (Art. 64); therefore the origin of coordinates is dx2 a cusp of the second species. Ex. 2. Let the curve be the semicubical parabola whose equation is a (y — b)2 = x3. b, x = 0, and from the equation of the curve we have dx dy to the axis of x at the distance b. The values of dx* have different signs, and therefore the cusp at the point x = species, as in the figure. Ex. 3. Let the equation to the curve be a (y - b)2 = y = b ± √ = a Now by assigning to x a small value less than c, the value of y is imaginary, whether the value of x be positive or negative, and therefore the point x = 0, y = b, is an isolated point. Scholium. 76. The singular points of curves may always be determined in the following manner :-The abscissa to which a singular point corresponds will be indicated by considering in what case the differential coefficients of any order whatever become 0, or ∞, or The nature of the point 0° will be assigned (1). By examining what number of branches pass through the point, and whether they extend on both sides of it. (2). By determining the position of the tangents to the curve at that point. (3). By the direction of their curvature. TRACING OF CURVES. 77. It is sometimes necessary to trace the general form of a curve from its equation without actually calculating its exact dimensions, and the principles we have just established afford considerable assistance in the delineation of the curve. When we proceed to trace a curve from its equation, it is desirable to solve the equation with respect to one or other of the coordinates, when that can be done in a form which enables us to determine readily the values of one coordinate for different values of the other. Let y = fx be the equation when thus resolved, then we may proceed as follows: = = (1). Assign to x all positive values from 0 to oo, marking those which make y 0, y ∞, or y impossible. The first gives the points where the curve cuts the axis of x, the second gives the infinite brauches, and the third gives the limits of the curve in the plane of reference. (2). Assign to x all negative values from 0 to ∞, and proceed as in the case of the positive values of x, attending to the positive and negative values of y in both cases, so as to obtain the branches on both sides of the axis of x. (3). Ascertain whether the curve has asymptotes, and draw them if they exist. dy (4). Find the value of and thence determine the angles at dx' which the curve cuts the axis, as well as the maximum and minimum values of y, if there be such. d2 y (5). Find the value of and thence deduce the nature of the curvature of the different branches, and the points of contrary flexure, if such exist. (6). Determine the nature and situation of the singular points, if there be such, by the usual rules. (7). When the equation to a curve is given by an equation r = fe, the values of 0, which make fo O are then to be found, and these give the angles at which the branches of the curve which pass through the origin cut the axis. Find the values of r when the curve cuts the axis, by giving to the values 0 and n; and by giving to the value (2n+1), we determine the values of r when the radius vector is perpendicular to the axis. Lastly, by dr 0, we determine the values of 0, which render r d Ꮎ making a maximum or a minimum. EXAMPLES. 1. Analyse and trace the curve whose equation is x y2 +2 a2 y − x2 = 0 Solving the equation for y, we have y a2 = х + x2 co, since the equation takes the form Now if we take the upper sign, and expand the radical in ascending powers of x, we get When a is small, the first term determines the sign of the whole series, and therefore y is positive. Also since no value of x can make y = 0, this branch of the curve lies always in the positive region and extends to infinity, since y = ∞, when x = 0. If we take the lower sign, and expand the radical in descending powers which when x = ∞ is negative and infinite, that is y Expanding in ascending powers of x, we get which, when x=0, is negative and infinite; hence this branch lies wholly between the positive axis of x and the negative axis of y. Since the equation of the curve is unchanged by the substitution of - x and -y for x and y, it follows that the curve is symmetrical in the opposite regions of coordinates. To determine the asymptotes, we have (Art. 53) +1); 814 ။ a2 = ∞ = 0. which, when x approaches to, gives ẞ Hence the equations of the asymptotes are y = ±x, and we have already seen that when y then x = 0; therefore the axis of y is also an asymptote. The asymptotes ZO Z' and WO W' bisect the angles between the axes of coordinates. Differentiating the equation of the curve we get dy 3x2 = y3 dx 2 (a* + xy) 0; hence the axis of x touches the curve at the origin of co- X' ordinates. From the last equation 3 x2 = 4 y2 0; hence y=x/3, and by substitution in the given equation we have 22 a2 x √3 = 0, which gives x = ± a* √√3; hence the minimum value of y belongs only to the branches in the second and fourth quadrants, the negative value of y having been employed for the fourth quadrant. Differentiating the equation of the curve twice, we get 2 (a2 + xy)2 X which, when x = 0, y = 0, gives the origin is a point of contrary flexure, which is otherwise evident, since the curve at the origin both touches and cuts the axis of x. If the curve has any other points of inflexion, they will be determined by putting the numerator of the last equation equal to zero, and solving the resulting equation. 2. Analyse and trace the curve whose equati on is Here r is equal to 0, when sin 0 = 0, and cos 0 or when becomes infinite, the curve consists of four loops arranged round the pole O. VOL. I. 2 c |
