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 — bg2 — ch3 = 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 8 meets the axes?

Here the axes x = 0, y = 0, are chords of intersection, and the equation must be of the form S = kry, 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

251. The conics S, S-kaß will touch; or, in other words, two of their points of intersection will coincide; if either a or B 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 ST (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

H H.

equation, and that the coefficient of x 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 zy. If S' be a circle, the axes of SkS' 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 Skaß, 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 λ =

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

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= 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 the 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 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 xyk (Art. 199) is a particular case of the form ay=B 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 y=px is also a particular case of ay=62. Writing the equation x (0.x+0.y+p)=y*, the form of the equation shows, not only that the line 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)* + (2gx + 2fy +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).

Ex. The general equation

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0

may be regarded as a particular case of the form (Art. 122) ay = kß8. For the first three terms denote two lines a, y passing through the origin, and the last three terms denote the line at infinity ß, together with the line 8, 2gx+2y+c. The form of the equation then shows that the lines a, y meet the curve at infinity, and also that represents the line joining the finite points in which ay meet the curve.

235. In accordance with Art. 253, the equation S= kB is to be regarded as a particular case of Sas, and denotes a system of conics passing through the two finite points where 8 meets 8, and also through the two infinitely distant points where S is met by Qe+Q#+4. Now it is plain that the coefficients of **, ofey, and of y', are the same in S and in S-ks, and therefire (Art. 231) that these equations denote conics similar and similarly placed. We learn, therefore, that two conics similar and similarly placed meet each other in two infinitely distant points, and consequently only in two finite points.

Y

This is also goometrically evident when the curves are hyperbolas; for the asymptotes of similar conics are parallel (Art. 233), that is, they intersect at infuity; but each asymptote intersects its own curve at infinity; consequently the infinitely distant point of intersection of the two parallel asymptotes is also a point common to the two curves. Thus, on the figure, the infinitely distant point of meeting of the lines OX, Ox,

X

One of

and of the lines OY, Oy, are common to the curves. their finite points of intersection is shown on the figure, the other is on the opposite branches of the hyperbolas.

If the curves be ellipses, the only difference is that the asymptotes are imaginary instead of being real. The directions of the points at infinity, on two similar ellipses, are determined from the same equation (ax+2hxy + by2 = 0) (Arts. 136, 234). Now, although the roots of this equation are imaginary, yet they are, in both cases, the same imaginary roots, and therefore the curves are to be considered as having two imaginary points at infinity common. In fact, it was observed before, that even when the line a does not meet S in real points, it is to be re

garded as a chord of imaginary intersection of S and S-kaß, and this remains true when the line a is infinitely distant.

If the curves be parabolas, they are both touched by the line at infinity (Art. 254); but the direction of the point of contact, depending only on the first three terms of the equation, is the same for both. Hence, two similar and similarly placed parabolas touch each other at infinity. In short, the two infinitely distant points common to two similar conics are real, imaginary, or coincident, according as the curves are hyperbolas, ellipses, or parabolas.

256. The equation S=k, or S=k(0.x +0.y +1)' is manifestly a particular case of S=ka, and therefore (Art. 252) denotes a conic having double contact with S, the chord of contact being at infinity. Now S-k differs from S only in the constant term. Not only then are the conics similar and similarly placed, the first three terms being the same, but they are also concentric. For the coordinates of the centre (Art. 140) do not involve c, and therefore two conics whose equations differ only in the absolute term are concentric (see also Art. 81). Hence, two similar and concentric conics are to be regarded as touching each other at two infinitely distant points. In fact, the asymptotes of two such conics are not only parallel but coincident; they have therefore not only two points at infinity common, but also the tangents at those points; that is to say, the curves touch.

If the curves be parabolas, then, since the line at infinity touches both curves, S and S-k have with each other, by Art. 251, a contact at infinity of the third order. Two parabolas whose equations differ only in the constant term will be equal to each other; for the curves y=px, y=p (x+n) are obviously equal, and the equations transformed to any new axes. will continue to differ only in the constant term. We have seen, too (Art. 205), that the expression for the parameter of a parabola does not involve the absolute term. The parabolas then, S and S-k are equal, and we learn that two equal and similarly placed parabolas whose axes are coincident may be considered as having with each other a contact of the third order at infinity.

257. All circles are similar curves, the terms of the second degree being the same in all. It follows then, from the last