condition that the circle should osculate is -g g=-rb sin w, or r= b sin w The quantity r is called the radius of curvature of the conic at the point T. 242. To find the radius of curvature at any point on a central conic. In order to apply the formula of the last Article the tangent at the point must be made the axis of y. Now the equation referred to a diameter through the point and its conjugate (+1) is transferred to parallel axes through the given point, by substituting x+a' for x, and becomes a'2 Therefore, by the last a' sin w 2 y" 2x Article, the radius of curvature is Now a' sin is the perpendicular from the centre on the tangent, therefore the radius of curvature. 243. Let N denote the length of the normal PN, and let y denote the angle FPN between the normal and focal radius vector, then the radius of Q b b' =- (Art. 188), whence the truth of the formula is *In the Examples which follow we find the absolute magnitude of the radius of curvature, without regard to sign. The sign, as usual, indicates the direction in which the radius is measured. For it indicates whether the given curve is osculated by a circle whose equation is of the form x2 + 2xy cos w + y2 2rx sin ∞ = 0, the upper sign signifying one whose centre is in the positive direction of the axis of x; and the lower, one whose centre is in the negative direction. The formula in the text then gives a positive radius of curvature when the concavity of the curve is turned in the positive direction of the axis of x, and a negative radius when it is turned in the opposite direction. Thus we have the following construction: Erect a perpendicular to the normal at the point where it meets the axis; and again at the point Q, where this perpendicular meets the focal radius, draw CQ perpendicular to it, then C will be the centre of curvature, and CP the radius of curvature. 244. Another useful construction is founded on the principle that if a circle intersect a conic, its chords of intersection will make equal angles with the axis. For the rectangles under the segments of the chords are equal (Euc. III. 35), and therefore the parallel diameters of the conic are equal (Art. 149), and therefore make equal angles with the axis (Art. 162). Now, in the case of the circle of curvature, the tangent at T (see figure, p. 226) is one chord of intersection and the line TL the other; we have, therefore, only to draw TL, making the same angle with the axis as the tangent, and we have the point L; then the circle described through the points T, L, and, touching the conic at T, is the circle of curvature. This construction shows that the osculating circle at either vertex has a contact of the third degree. Ex. 1. Using the notation of the eccentric angle, find the condition that four points a, ß, y, & should lie on the same circle (Joachimsthal, Crelle, XXXVI. 95). The chord joining two of them must make the same angle with one side of the axis as the chord joining the other two does with the other; and the chords being we have tan (a + ẞ) + tan§ (y + 8) = 0 ; a + ß + y + ô = 0; oг = 2mπ. Ex. 2. Find the coordinates of the point where the osculating circle meets the conic again. We have a = ẞ= y; hence 8=- 3a; or X= 42'3 a2 4y's Ex. 3. If the normals at three points a, ẞ, y meet in a point, the foot of the fourth normal from that point is given by the equation a + B + y + d = (2m + 1) π. Ex. 4. Find the equation of the chord of curvature TL. Ex. 5. There are three points on a conic whose osculating circles pass through a given point on the curve; these lie on a circle passing through the point, and form a triangle of which the centre of the curve is the intersection of bisectors of sides (Steiner, Crelle, XXXII. 300; Joachimsthal, Crelle, XXXVI. 95). Here we are given d, the point where the circle meets the curve again, and fre the last Example the point of contact is a=d. But since the sine and co of would not alter if d were increased by 360°, we might also have a = - -fo + 120°, or=- jo+240°, and, from Ex. 1, these three points lie on a circle passing through ò. If in the last Example we suppose X, Y given, since the cubics which determine randy' want the second terms, the sums of the three values of x' and of y' are respectively equal to nothing; and therefore (Ex. 4, p. 5) the origin is the intersection of the bisectors of sides of the triangle formed by the three points. It is easy to see that when the bisectors of sides of an inscribed triangle intersect in the centre, the normals at the vertices are the three perpendiculars of this triangle, and therefore meet in a point. 245. To find the radius of curvature of a parabola. The equation referred to any diameter and tangent being y=p'x, the radius of curvature (Art. 241) is is the angle between the axes. The expression p' where and the N cos'y construction depending on it, hold for the parabola, since Ex. 1. In all the conic sections the radius of curvature is equal to the cube of the normal divided by the square of the semi-parameter. Ex. 2. Express the radius of curvature of an ellipse in terms of the angle which the normal makes with the axis. Ex. 3. Find the lengths of the chords of the circle of curvature which pass through the centre or the focus of a central conic section. 26'2 26'2 Ans. and a'' α Ex. 4. The focal chord of curvature of any conic is equal to the focal chord of the conic drawn parallel to the tangent at the point. Ex. 5. In the parabola the focal chord of curvature is equal to the parameter of the diameter passing through the point. 246. To find the coordinates of the centre of curvature of a central conic. These are evidently found by subtracting from the coordinates of the point on the conic the projections of the radius of curvature upon each axis. Now it is plain that this radius is to its projection on y as the normal to the ordinate y. We find the projection, therefore, of the radius of curvature on the axis of We should have got the same values by making a=B=y in Ex. 8, p. 220. Or, again, the centre of the circle circumscribing a triangle is the intersection of perpendiculars to the sides at their middle points; and when the triangle is formed by three consecutive points on a curve, two sides are consecutive tangents to the curve, and the perpendiculars to them are the corresponding normals, and the centre of curvature of any curve is the intersection of two consecutive normals. Now if we make x' = x" = X, y=y" Y, in Ex. 4, p. 175, we obtain again the same values as those just determined. = 247. To find the coordinates of the centre of curvature of a parabola. The projection of the radius on the axis of y is found in like manner (by multiplying the radius of curvature = Y’ sin'ei and subtracting this quantity from y' we have 4y's p* (Art. 212). N sin'0 y' N Р The same values may be found from Ex. 10, p. 214. 248. The evolute of a curve is the locus of the centres of curvature of its different points. If it were required to find the evolute of a central conic, we should solve for x'y' in terms of the x and y of the centre of curvature, and, substituting in the c2 equation of the curve, should have (writing = = A, a In like manner the equation of the evolute of a parabola is found to be which represents a curve called the semi-cubical parabola. CHAPTER XIV. METHODS OF ABRIDGED NOTATION. 249. IF S=0, S'0 be the equations of two conics, then the equation of any conic passing through their four, real or imaginary, points of intersection can be expressed in the form SkS'. For the form of this equation shows (Art. 40) that it denotes a conic passing through the four points common to S and S'; and we can evidently determine k so that SkS. shall be satisfied by the coordinates of any fifth point. It must then denote the conic determined by the five points.* This will, of course, still be true if either or both the quantities S, S' be resolvable into factors. Thus Skaß, being evidently satisfied by the coordinates of the points where the right lines a, ẞ meet S, represents a conic passing through the four points where S is met by this pair of lines; or, in other words, represents a conic having a and B for a pair of chords of intersection with S. If either a or B do not meet S in real "points, it must still be considered as a chord of imaginary intersection, and will preserve many important properties in relation to the two curves, as we have already seen in the case of the circle (Art. 106). So, again, ay=kßd denotes a conic circumscribing the quadrilateral aßyd, as we have already seen (Art. 122). It is obvious that in what is here stated, a need not *Since five conditions determine a conic, it is evident that the most general equation of a conic satisfying four conditions must contain one independent constant, whose value remains undetermined until a fifth condition is given. In like manner, the most general equation of a conic satisfying three conditions contains two independent constants, and so on. Compare the equations of a conic passing through three points or touching three lines (Arts. 124, 129). If we are given any four conditions, in the expression of each of which the coefficients enter only in the first degree, the conic passes through four fixed points; for by eliminating all the coefficients but one, the equation of the conic is reduced to the form S = kS'. † If aß be one pair of chords joining four points on a conic S, and yo another pair of chords, it is immaterial whether the general equation of a conic passing through the four points be expressed in any of the forms S - kaß, S – kyd, aß – kyô, where k is indeterminate; because, in virtue of the general principle, S is itself of the form «β - Αγ. |