that "if two right lines, drawn through the point of contact of two conics, meet the curves in points P, p, Q, q, then the chords PQ, pq, will meet on the chord of intersection of the conics."

This is a particular case of a theorem given in Art. 265, since one intersection of common tangents to two conics which touch, reduces to the point of contact (Art. 123).

=

268. The equation of a conic circumscribing a quadrilateral (ay kẞ8) furnishes us with a proof of "Pascal's theorem," that the three intersections of the opposite sides of any hexagon inscribed in a conic section are in one right line.

Let the vertices be abcdef, and let ab = 0 denote the equation of the line joining the points a, b, then, since the conic circumscribes the quadrilateral abcd, its equation must be capable of being put into the form ab. cd - bc. ad = 0.

But since it also circumscribes the quadrilateral defa, the same equation must be capable of being expressed in the form de. fa- ef. ad = 0.

From the identity of these expressions we have

ab.cd - de. fa= (bc – ef) ad.

Hence we learn that the left-hand side of this equation (which from its form represents a figure circumscribing the quadrilateral formed by the lines ab, de, cd, af) is resolvable into two factors, which must therefore represent the diagonals of that quadrilateral. But ad is evidently the diagonal which joins the vertices a and d, therefore bc - ef must be the other, and must join the points (ab, de), (cd, af); and since from its form it denotes a line through the point (bc, ef), it follows that these three points are in one right line. We shall in the next chapter give another demonstration of this important theorem.

By supposing two vertices of the hexagon to be indefinitely near, we may, "given five points on a conic, draw a tangent at any of these points."

269. We may, as in the case of Brianchon's theorem, obtain a number of different theorems concerning the same six points, according to the different orders in which we take them. Thus

since the conic circumscribes the quadrilateral beef, its equation can be expressed in the form

be.cf - bc.ef= 0.

Now, from identifying this with the first form given in the last Article, we have

whence, as before, we learn that the three points (ab, cf), (cd, be), (ad, ef) lie in one right line, viz. ad ef = 0.

In like manner, from identifying the second and third forms of the equation of the conic, we learn that the three points (de, cf), (fa, be), (ad, bc) lie in one right line, viz. bc – ad = 0. But the three right lines

bc - ef = 0, ef - ad = 0, ad - bc = 0,

meet in a point (Art. 37). Hence we have Steiner's theorem, that "the three Pascal's lines which are obtained by taking the vertices in the orders respectively, abcdef, adcfeb, afcbed, meet in a point." For some further developments on this subject we refer the reader to the note at the end of the volume.

TRILINEAR CO-ORDINATES.

270. We proved (Art. 61) that being given three lines (a, ẞ, y), we can express the equation of any other right line in the form Aa + BB + Cy = 0.

In the same manner we can show that there is no conic section whose equation may not be written in the form

Aa2 + Baß + Cẞß2 + Day + Eẞy + Fy2 = 0.

For this equation is obviously of the second degree ; and since it contains five independent constants, we may (as in Art. 128) determine these constants so that the curve which it represents may pass through five given points, and therefore may coincide with any given conic. In short, since the equation just written contains the same number of constants as the equation

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0,

it must be equally capable of representing any particular conic. In like manner, in general, there is no curve of any degree

whose equation may not be expressed as a homogeneous function of the quantities a, ß, y. For it can readily be proved that the number of terms in the complete equation of the nth order between two variables is the same as the number of terms in the homogeneous equation of the nth order between three variables.

271. If (as in Art. 66) we render the Cartesian equation homogeneous by the introduction of the linear unit z, we at once perceive the identity of the two forms

Aa2 + Baß + Cẞ2 + Day + Eẞy + Fy2 = 0,

Ax2 + Bxy + Cy2 + Dxz + Eyz + Fz2

=

0;

the latter being the form assumed by the former, when two of the lines of reference (aß) are the axes (xy), and the third (y) is the line at infinity z. It is important to keep constantly in view the analogy which subsists between these two forms of equations. If, for instance, we make y = 0 in the first equation, the result Aa2 + Baß + Cẞ2 = 0 is plainly the equation of the lines joining the point (aß) to the points where y cuts the curve. In like manner, if we make 20 in the second equation, the result Ax2 + Bxy + Cy2 = 0 must be the equation of the pair of lines joining the origin (xy) to the points where the line at infinity cuts the curve (Art. 134).

z =

Precisely the same argument which proves (Art. 36) that the curve represented by

(Aa2 + Baß + Cß2) + y (Da + Eß + Fy) = 0

passes through the intersections of the line y with the pair of lines (Aa2 + Baẞ+ Cẞ2), proves likewise that the curve passes through the intersections of the same pair of lines with the line Da + Eẞ + Fy = 0. This latter equation then denotes the fourth side of a quadrilateral inscribed in the conic, of which the other three sides are the line y, and the lines joining to aẞ the points where meets the curve. γ In like manner Dx + Ey + F = 0 is the equation of a line joining the two finite points where the curve is met by two lines drawn through the origin to meet the curve at infinity.

In general let the equation of a curve of any degree be written

Un + Un-1Z + Un-222 + Un-323 + &c. = 0,

(where we use the abbreviations un, un-1, &c. to denote terms of the nth, n − 1st, &c. degrees). Now, if we seek the points where the line at infinity meets the curve, we have only to make z = 0, when we obtain the equation u2 = 0; hence we infer that the directions of the points at infinity on any curve are found by putting the highest terms of the equation = 0.

un

Again, we saw (Art. 136), that, if A = 0 in the equation of the second degree, the axis of x will meet the curve in one infinitely distant point. The same thing appears, by making y = 0 in the equation, which will then reduce to

The axis, therefore, meets the curve, not only in the finite point where it meets the line (Dx + F), but also in the point at infinity where it meets the line z.

In like manner, if both A and D = 0, the points where the axis meets the curve are given by the equation F22 = 0; hence, the axis meets the curve in two coincident points at infinity, and is, therefore, an asymptote.

272. We shall commence our examples of the use of trilinear co-ordinates with the equation (Art. 254) of a conic section, referred to two tangents and their chord of contact,

LM = R2,

and shall first show how to express the equation of any line connected with the conic in terms of L, M, R.

We can express the position of any point on the curve by a single variable (Art. 234); for if μL = R be the equation of the line joining any point on the curve to (LR), then, substituting in the equation of the curve, we get

M = μR and μ3L = M

for the equations of the lines joining this point to (MR) and (LM): any two of these three equations, therefore, will determine a point on the curve. We shall call this point the point μ.

We can form, by Art. 59, the equation of the line joining two points on the curve u and u', and we get

an equation evidently satisfied by either of the suppositions (μL = R, μR = M), or (u'L= R, μ'R = M).

If μ and u' coincide, we find the equation of the tangent, viz., μL 2μR+ M = 0.

Hence, conversely, if the equation of a right line (μ3L-2μR+M=0) contain an indeterminate quantity u in the second degree, the right line will always touch a conic section (LM = R3).

273. Given four points of a conic, the anharmonic ratio of the pencil joining them to any fifth point is constant. The lines joining four points μ, u", μ", μ"

to any fifth point

μ” (μL – R) + (M - μR) = 0,

μ”" (μL – R) + (M - μR) = 0,

and is, therefore, independent of the position of the point μ. We shall, for brevity, use the expression, "the anharmonic ratio of four points of a conic," when we mean the anharmonic ratio of a pencil joining those points to any fifth point on the

curve.

274. Four fixed tangents cut any fifth in points whose anharmonic ratio is constant.

Let the fixed tangents be those at the points u', u", μ", μ""'; and the variable tangent that at the point μ; then the anharmonic ratio in question is the same as that of the pencil joining the four points of intersection to the point LM. Now if we eliminate R from the equations of any two tangents,

the equation of the line joining LM to the intersection of these two tangents. The anharmonic ratio in question is therefore that of the four lines,

μμ L - Μ = 0, μμL - Μ = 0, μμ"L - Μ = 0, μμ"L - Μ = 0, which by Art. 55 is