Thus, for example, the equations of two conics, both passing through the origin, and having the line x for a common tangent are, (Art. 144) ax2 + 2hxy + by2+2gx=0; a'x2 + 2h'xy + b'y2 + 2g'x = 0. And, as in Ex. 2, p. 170, x {(ab' — a'b) x + 2 (hb' — h’b) y + 2 (gb' — g'b)} = 0, represents a figure passing through their four points of intersection. The first factor represents the tangent which passes through the two coincident points of intersection, and the second factor denotes the line LM passing through the other two points. If now gb'=g'b, LM passes through the origin, and the conics have contact of the second order. If in addition hb'=h'b, the equation of LM reduces to x=0; LM coincides with the tangent, and the conics have contact of the third order. In this last case, if we make by multiplication, the coefficients of y the same in both the equations, the coefficients of xy and x will also be the same, and the equations of the two conics may be reduced to the form ax2 + 2hxy + by3 +2gx=0, a'x2+2hxy + by2+2gx = 0. 240. Two conics may have double contact, if the points of intersection 1, 2 coincide and also the points 3, 4. The condition that the pair of conics, considered in the last article, should touch at a second point, is found by expressing the condition that the line LM, whose equation is there given, should touch either conic. Or, more simply, as follows: Multiply the equations by g' and g respectively, and subtract, and we get (ag' — a'g) x2 + 2 (hg' — h'g) xy + (bg' — b'g) y2 = 0, which denotes the pair of lines joining the origin to the two points in which LM meets the conics. And these lines will coincide if (ag' — a'g) (bg' — b'g) = (hg' — h'g)2. 241. Since a conic can be found to satisfy any five conditions (Art. 133), a conic can be found to touch a given conic at a given point, and satisfy any three other conditions. If it have contact of the second order at the given point, it can be made to satisfy two other conditions; and if it have contact of the third order, it can be made to satisfy one other condition. Thus we can determine a parabola having contact of the third order at the origin with ax2+2hxy + by2+2gx=0. Referring to the last two equations (Art. 239), we see that it is only necessary to write a' instead of a, where a' is determined by the equation a'b = h3. We cannot, in general, describe a circle to have contact of the third order with a given conic, because two conditions must be fulfilled in order that an equation should represent a circle; or, in other words, we cannot describe a circle through four consecutive points on a conic, since three points are sufficient to determine a circle. We can however easily find the equation of the circle passing through three consecutive points on the curve. circle is called the osculating circle, or the circle of curvature. This The equation of the conic to oblique or rectangular axes, being, as before, ax*+2hxy+by+2gx = 0, that of any circle touching it at the origin is (Art. 84, Ex. 3) x2+2xy cos w + y2 − 2rx sin∞ = 0. Applying the condition gb'g'b (Art. 239), we see that the condition that the circle should osculate is The quantity is called the radius of curvature of the conic at the point 7. 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 * 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 w = 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. equation referred to a diameter through the point and its conjugate (a 12 + b12 = 1), is transferred to parallel axes through the given point, by substituting x + a' for x, and becomes Therefore, by the last Article, the radius of curvature is a' sin w Now a' sino 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 P Q (Art. 188), whence the truth of the formula is 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. 214) 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. FF 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 + d) = 0 ; a + ß + y + d = 0; oг= 2mπ. Ex. 2. Find the co-ordinates of the point where the osculating circle meets the conic again. We have a = B= = y; hence - 3a; or X = 4.x'3 Ex. 3. 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 from the last Example the point of contact is a = - -38. But since the sine and cosine of ♪ would not alter if ♪ were increased by 360°, we might also have a = − fò + 120o, or = − 36+ 240°, and from Ex. 1, these three points lie on a circle passing through d. If in the last Example we suppose X, Y given, since the cubics which determine x' and y' 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 p' where 2 sin e N and the 2 cos' is the angle between the axes. The expression construction depending on it, hold for the parabola, since N=p' sin @ (Arts. 212, 213) and y = 90° — 0 (Art. 217). 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'' 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 co-ordinates of the centre of curvature of central conic. These are evidently found by subtracting from the co-ordinates 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=ß=y in Ex. 8, p. 209. 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. 170, we obtain again the same values as those just determined. 247. To find the co-ordinates 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 N y' sin20 N |