and in these points only. Hence since two conics in general intersect in four points, it follows that in this case the four points of intersection coincide two and two, that is, the conics. touch one another at the two points where they meet the above-mentioned line, or have double contact with each other.

The arbitrary constant k may be determined by making the conic pass through any assigned point. Suppose now that the conic is required to pass through the point (f, g, h), of which the line of contact is the polar. This gives, for the determination of k, the condition

$ (f, g, h) + k {(uf + w'g + v'h) ƒ + (w'f + vg + u'h) g

† (f, g, h) & (a, B, y) − {(uf + w'g+ v'h) a

+(w'f+vg + u'h) B + (v'f + u'g + wh) y}2 = 0, represents the curve of the second degree, passing through the point (f, g, h) and touching the conic (a, B, y) = 0, at the points where the polar of this point intersects it. But this curve must evidently be coincident with the two tangents drawn from that point to the given conic & (a, ß, y) = 0.

This equation may be put under another form, for the coefficient of a will be found, by actual expansion, to be u (uf2 + vg2 + wh2 + 2u'gh + 2v'hf+2w'fg)

=

− (u2ƒ2 + w22g2 + v'2h2 + 2v'w'gh + 2uv'hƒ + 2uw'fg)

= (uv — w'2) g2 + (wu — v'2) h2 + 2 (uu' — v'w') gh

= Wg+ Vh2-2U'gh.

That of 2ẞy is

u' (uf2 + vg2 + wh2 + 2u'gh + 2v'hf + 2w'fg)

=

= (uu' — v'w') ƒ2 + (u'2 — vw) gh+ (u'v' — uw') hf+ (w'u' — vv') fg

- U'ƒ2 — Ugh + W'hf+ V'fg.

Similar expressions holding for the coefficients of 2, y2, 2ya, 2aß, we obtain the equation of the two tangents under the form

2

(Wg2+Vh2-2U'gh)a2+(Uh2+Wƒ2−2V'hƒ)ß2+(Vƒ2+Ug2—2W'fg) y2

− 2 (U'ƒ2 +Ugh — W'hf-V'fg) By

− 2 (V'g2 + Vhf - Ufg -W'gh) ya

- 2 (W'h2+Wfg - V'gh — U'hf) aß =0.

If the point (f, g, h) be within the conic, these two tangents will be imaginary.

11. To find the co-ordinates of the centre.

Since the two tangents, drawn at the extremities of any chord passing through the centre, are parallel to each other, it follows that the pole of the centre is at an infinite distance, and may therefore be represented by the equation

Hence, if a, B, 7, be the co-ordinates of the centre, we obtain, by an investigation similar to that of Art. 9,

12. To find the equation of the asymptotes.

Writing a, B, y, for f, g, h, in the investigation of Art. 10, and paying regard to equations (A) of Art. 11, the asymptotes will be found to be represented by the equation

$ (ã, B, Ÿ) & (α, ß, y) − {(aa +bB+cy)2} k2 = 0,

But, multiplying equations (A) in order by a, B, 7, and adding, we get (a, B, 7)+2A. k = 0.

tion

or

Hence the asymptotes may be represented by the equa

$ (α, B, y) - ¢ (ã, B, y) = 0,

$ (a, B, y) + 2▲ . k = 0,

which may be put under the homogeneous form

(aã +bB+cy) & (a, B, y) + k (aa+bB+cy)2 = 0.

But, by the final result of Art. 11, it may be seen that aa+bB + cy Ua2+Vb2+Wc2 + 2 U′bc + 2 V'ca + 2 W'ab k

=

12

uvw + 2u'v'w' — uu2 vv ww'2

whence the equation of the asymptotes becomes

--

(Ua2 + Vb2 + Wc2 + 2 U'bc + 2 V'ca + 2 W'ab) & (a, ß, v)

w', v' (aa+bB+cy)2 = 0.

w', v,

u'

v', u', w

COR. It appears, from the preceding investigation, that if a, B, be the co-ordinates of the centre of the conic represented by the equation

Υ

$ (a, B, y) = ua2 + vß2 + wy2 + 2u'By + 2v'ya + 2w'aß = 0,

13. To find the condition that the conic may be a rectangular hyperbola.

If the equations of the asymptotes be

la + mB + ny = 0,

l'a + m'ß + n'y = 0,

the condition of their perpendicularity is

ll' + mm' + nn' — (mn' + m'n) cos A — (nl' + n'l) cos B

Ua2 + Vb2 + Wc2 + 2 U'bc + 2 V'ca + 2 W'ab = D,

uvw + Qu'v'w'. uu'2 · vv'2 — ww'2 = K,

we see, by reference to Art. 12, that

=

1 (mn' + m'n) ___ — (nl' + n' l ) ___ 1 (lm' + I'm)

Du'-Kbc

=

=

Do'- Kca Dw' - Kab

Hence the required condition is

D (u+v+w-Qu' cos A-2v' cos B-2w' cos C)

− K (a2 + b2 + c2 − 2bc cos A – 2ca cos B-2ab cos C) = 0.

Now a2+b2+ c2 - 2bc cos A-2ca cos B-2ab cos C=0 identically, hence the required condition becomes

u+v+w — Qu' cos A - 2v' cos B-2w' cos C = 0.

COR. It hence appears, that the condition that the

conic

u'ẞy + v'ya + w'aß = 0,

described about the triangle of reference, may be a rectangular hyperbola, is

u'cos A+ v' cos B+w' cos C = 0;

that is, the conic must pass through the point determined by the equations

a cos A = ẞ cos By cos C.

This point (see Art. 5, Chap. 1.) is the point of intersection of the perpendiculars let fall from each angular point of the triangle on the opposite side. Hence we obtain the following elegant geometrical proposition, that every rectangular hyperbola described about a given triangle passes through the point of intersection of the perpendiculars let fall from each angular point of the triangle on the opposite side.

Again, if u', v', w' be all = 0, the condition is

u + v + w = 0,

proving that, if the equation

ua2 + vß3 + wy2 = 0

represent a rectangular hyperbola, the curve will pass through the four points for which

a=B=Y, - a=B=y, α=- -B=y, α=ß = −y.