Y= and subtracting this quantity from y', we have 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 equa tion of the curve, should have (writing c2 α A, c2 = B), In like manner the equation of the evolute of a parabola is found 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 co-ordinates 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 co-ordinates 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 ẞ 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 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 – kyô, aß – kyồ, where k is indeterminate; because, in virtue of the general principle, S′ is itself of the form αβ - Αγδ. 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-af-bg" — 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", 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 the chords of intersection, and the equation must be of the form S = kxy, where k is indeterminate. See Ex. 1, p. 148. 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 co-ordinates of the fifth point (-4, 3), we obtain k = — 223. 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 ß 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 S=T(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 = equation, and that the coefficient of x that of y'; 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=kT2. 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 co-ordinates 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 co-ordinates 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 And expressing that the coefficient of x2 = that of y2, we find the equation becomes = 6'2 b2 and 2 (α2 — b2) x13x 2 (b2 — a2) y'3y +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 − 3px'}. 252. We have seen that Skaß represents a conic passing α and Q to q. Suppose then that the lines a and ẞ 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=ka* 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 an imaginary chord of contact of the conics S and S-ka". 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 form S-kTT", where T and T" represent the tangents at these points. 253. If the line a be parallel to an asymptote of the conic S, it will also be parallel to an asymptote of any conic represented by Skaß, which then denotes a system passing through three finite, and one infinitely distant point. In like manner, if in addition ẞ were parallel to the other asymptote, the system would pass through two finite and two infinitely distant points. Other forms which denote conics having points of intersection at infinity, will be recognized by bearing in mind the principle (Art. 67) that the equation of an infinitely distant line is 0.x+0.y+C=0; and hence (Art. 69) that an equation, apparently not homogeneous, may be made homogeneous in form, if in any of the terms which seem to be below the proper degree of the equation we replace one or more of the constant multipliers by 0.x +0.y +C. Thus, the equation of a conic referred to its asymptotes xy=k2 (Art. 199), is a particular case of the form ay=ß2 referred to two tangents and the chord of contact (Arts. 123, 252). Writing the equation xy=(0.x+0.y+k)2, it is evident that the lines x and y are tangents, whose points of contact are at infinity (Art. 154). 254. Again, the equation of a parabola y2=px is also a particular case of ay=B". Writing the equation x(0.+0.y+p)=y2; the form of the equation shows not only that the line x touches the curve, its point of contact being the point where x meets y, but also that the line at infinity touches the curve, its point of contact also being on the line y. The same inference may be drawn from the general equation of the parabola (ax+ By)2 + (2gx+2ƒy + c) (0.x + 0. y + 1) = 0, which shews that both 2gx+2fy+c, and the line at infinity are tangents, and that the diameter ax + By joins the points of contact. Thus, then, every parabola has one tangent altogether at an infinite distance. In fact, the equation which determines the direction of the points at infinity on a parabola is a perfect square (Art. 137); the two points of the curve at infinity therefore coincide; and therefore the line at infinity is to be regarded as a tangent (Art. 83). |