or of d would not alter if & were increased by 360°, we might also have a = — - 3+ 120°, 30+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 ' 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 and the N is the angle between the axes. The expression construction depending on it, hold for the parabola, since N=p' sin 0 (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. a'' 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 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 b''y' y (by multiplying the radius by 2) = 577. p N The y of the c2 y'. But b=6"+", there fore the y of the centre of curvature is. In like We should have got the same values by making a=ẞ=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 N sin20 = 3x' + p. 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 equation of the curve, should have (writing c2 c2 - = · A, % = B), 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 S=kS. 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 S-kaß, 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 ẞ 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=kß8 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 SkS'. † If aß be one pair of chords joining four points on a conic S, and yd 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 – kyd, where k is indeterminate; because, in virtue of the general principle, S is itself of the form αβ -- λγδ. be restricted, as at p. 53, to denote a line whose equation has been reduced to the form x cosa + y sin a=p; but that the argument holds if a denote a line expressed by the general equation. 250. There are three values of k, for which S-kS' represents a pair of right lines. For the condition that this shall be the case, is found by substituting a- ka', b-kb', &c. for a, b, &c. in abc+2fgh — af2 — bg3 — ch2 = 0, and the result evidently is of the third degree in k, and is therefore satisfied by three values of k. If the roots of this cubic be k', k", "", then S-k'S', S-k"S', S-k" S', denote the three pairs of chords joining the four points of intersection of S and S' (Art. 238). Ex. 1. What is the equation of a conic passing through the points where a given conic S meets the axes? Here the axes x = 0, y = 0, are chords of intersection, and the equation must be of the form Skxy, where k is indeterminate. See Ex. 1, Art. 151. Ex. 2. Form the equation of the conic passing through five given points; for example (1, 2), (3, 5), (— 1, 4), (— 3, − 1) (— 4, 3). Forming the equations of the sides of the quadrilateral formed by the first four points, we see that the equation of the required conic must be of the form (3x - 2y+1) (5x − 2y + 13) = k (x − 4y + 17) (3x − 4y + 5). Substituting in this, the coordinates of the fifth point (−4,3), we obtain k=-W. Substituting this value and reducing the equation, it becomes 79x2 - 320xу + 301y2 + 1101x — 1665y + 1586 = 0. 251. The conics S, S-kaß will touch; or, in other words, two of their points of intersection will coincide; if either a or ß touch S, or again, if a and 8 intersect in a point on S. Thus if T=0 be the equation of the tangent to S at a given point on it x'y', then S T (lx + my + n), is the most general equation of a conic touching S at the point x'y'; and if three additional conditions are given, we can complete the determination of the conic by finding l, m, n. = Three of the points of intersection will coincide if lx+my+n pass through the point x'y'; and the most general equation of a conic osculating S at the point x'y' is ST (lx+my — lx'— my'). If it be required to find the equation of the osculating circle, we have only to express that the coefficient xy vanishes in this H H. 2 equation, and that the coefficient of x2= that of y2; when we have two equations which determine 7 and m. The conics will have four consecutive points common if lx+my+n coincide with T, so that the equation of the second conic is of the form S-kT. Compare Art. 239. Ex. 1. If the axes of S be parallel to those of S', so will also the axes of SkS'. For if the axes of coordinates be parallel to the axes of S, neither S nor S' will contain the term xy. If S' be a circle, the axes of S-kS' are parallel to the axes of S. If S-kS' represent a pair of right lines, its axes become the internal and external bisectors of the angles between them; and we have the theorem of Art. 244. Ex. 2. If the axes of coordinates be parallel to the axes of S, and also to those of S-kaß, then a and ẞ are of the forms lx + my + n, lx − my + n'. Ex. 3. To find the equation of the circle osculating a central conic. The equation must be of the form Expressing that the coefficient of xy vanishes, we reduce the equation to the form y2 (xx' Yy' yy' y'2 b2 a2 b2 + and expressing that the coefficient of x2= that of y2, we find λ = +a^2 - 26'2 = 0. Ex. 4. To find the equation of the circle osculating a parabola. - Ans. (p2+4px') (y2 — px) = {2yy' — p (x + x′)} {2yy' + px - 3рx'}, 252. We have seen that S-kaß represents a conic passing and Q to q. Suppose that the lines a and B coincide, then the points P,p; Q, q coincide, and the second conic will touch. the first at the points P, Q. Thus, then, the equation S=ka2 represents a conic having double contact with S, a being the chord of contact. Even if a do not meet S, it is to be regarded as the imaginary chord of contact of the conics S and S-ka2. In like manner ay kß represents a conic to which a and y are tangents and ẞ the chord of contact, as we have already seen (Art. 123). The equation of a conic having double contact with S at two given points x'y', x"y" may be also written in the = |