b

a

, then every line parallel to the axis minor will be unaltered

by projection, but every line parallel to the axis major will be shortened in the ratio ba; the projection will, therefore (Art. 163), be a circle, whose radius is b.

369. We shall apply the principles laid down in the last Article to investigate the expression for the radius of a circle circumscribing a triangle inscribed in a conic, given Ex. 7, p. 220.*

Let the sides of the triangle be a, B, y, and its area A, then, by elementary geometry,

Now let the ellipse be projected into a circle whose radius is b, then, since this is the circle circumscribing the projected triangle, we have

But, since parallel lines are in a constant ratio to their projections, we have

a:a::b: b',

B': B::b:b",

y: y :: b: b"";

and since (Art. 368) A' is to A as the area of the circle (=πb2) to the area of the ellipse (= Tab) (see chap. XIX.), we have

*This proof of Mr. Mac Cullagh's theorem is due to Dr. Graves.

CHAPTER XVIII.

INVARIANTS AND COVARIANTS OF SYSTEMS OF CONICS.

370. It was proved (Art. 250) that if S and S' represent two conics, there are three values of k for which kS+ S' represents a pair of right lines. Let

S=ax2 + by2 + cz2 +2fyz +2gzx +2hxy, S' = a'x2 + b'y2 + c'z2 + 2ƒ ̃yz + 2g′zx + 2h'xy. We also write

Δ ▲ = abc + 2fgh- af bg-ch',

D'a'b'c'+2fgh' - af" - b'g" - c'h2.

Then the values of k in question are got by substituting ka + a', kb+b', &c. for a, b, &c. in ▲ = 0. We shall write the resulting cubic Ak3 + Ok2 + O ́k + A′ = 0.

The value of C, found by actual calculation, is

+2 (gh − af) ƒ' + 2 (hf − bg) g′ + 2 (fg — ch) h' ;

or, using the notation of Art. 151,

Aa' + Bb'+Cc+2Ff' + 2 Gg' + 2Hh' ;

as is also evident from Taylor's theorem. The value of 'is got from by interchanging accented and unaccented letters, and may be written

✪' = A'a + Bb + C'c+2F'ƒ+2 G'g+2H’h.

If we eliminate k between kS+ S'=0, and the cubic which determines k, the result

AS" - OS"S+O'S'S' - A'S3 = 0,

(an equation evidently of the sixth degree), denotes the three pairs of lines which join the four points of intersection of the two conics (Art. 238).

Ex. To find the locus of the intersection of normals to a conic, at the extremities of a chord which passes through a given point aß. Let the curve be S =

x2

y2

1;

+ then the points whose normals pass through a given point x'y' are determined (Art. 181, Ex. 1) as the intersections of S with the hyperbola S′ = 2 (c2xy + b2y'x — a2x'y). We can then, by this article, form the equation of the six chords which join the feet of normals through x'y', and expressing that this equation is satisfied for the point aß, we have the locus required.

We have A = −

1 " a2f2

→ = 0, 0' = (a2x22 + b2y'2 — c1), ▲′ = — 2a2b2c2x'y'. The equation of the locus is then

—

(a2ßx − b3ay — c°aß)3 + 2 (a2x2 + b2y2 — c1) (a2ßx − b3ay — c3aß) (~7 +

+ 1)=0,

a2b

which represents a curve of the third degree. If the given point be on either axis, the locus reduces to a conic, as may be seen by making a = 0 in the preceding equation. It is also geometrically evident, that in this case the axis is part of the locus. The locus also reduces to a conic if the point be infinitely distant; that is to say, when the problem is to find the locus of the intersection of normals at the extremities of a chord parallel to a given line.

371. If on transforming to any new set of coordinates, Cartesian or trilinear, S and S become S and S, it is manifest that kS+S becomes kS+ S, and that the coefficient k is not affected. It follows that the values of k, for which S+ S' represents right lines, must be the same, no matter in what system of coordinates S and S are expressed. Hence, then, the ratio between any two coefficients in the cubic for k, found in the last Article, remains unaltered when we transform from any one set of coordinates to another. The quantities ▲, O, O', A' are on this account called invariants of the system of conics. If then, in the case of any two given conics, having by transformation brought S and S to their simplest form, and having calculated A, O, O', A', we find any homogeneous relation existing between them, we can predict that the same relation will exist between these quantities, no matter to what axes the equations are referred. It will be found possible to express in

* It may be proved by actual transformation that if in S and S' we substitute for x, y, z; lx + my + nz, l'x + m'y + n'z, l''x + m'y+n"z, the quantities ▲, O, O' A' for the transformed system, are equal to those for the old, respectively multiplied by the square of the determinant

terms of the same four quantities the condition that the conics should be connected by any relation, independent of the position of the axes, as is illustrated in the next Article.

The following exercises in calculating the invariants ▲, O, O', A', include some of the cases of most frequent occurrence.

Ex. 1. Calculate the invariants when the conics are referred to their common self-conjugate triangle. We may take

S = ax2 + by2 + cz2, S' = a'x2 + b'y2 + c'z2 ;

and we may further simplify the equations by writing x, y, z, instead of x √(a'), y (b'), z √(c'), so as to bring S' to the form 2 + y2+22. We have then

And S+kS' will represent right lines, if

k3 + k2 (a + b + c) + k (bc + ca + ab) + abc = = 0.

And it is otherwise evident that the three values for which S+kS' represents right lines are - a, b, — c.

Ex. 2. Let S', as before, be x2 + y2 + 22, and let S represent the general equation. Ans. ✪ = (bc — ƒ2) + (ca − g2) + (ab − h2) = A + B + C ; O' = a + b + c. Ex. 3. Let S and S' represent two circles x2 + y2 — r2, (x − a)2 + (y − (3)2 — go12 ̧ Ans. ▲ = r2, → = a2 + ß2 — 2r2 — r22, 0′ = a2 + ß32 — pr2 — 2p12, ▲′ = — go12 ̧ So that if D be the distance between the centres of the circles, S+kS' will represent right lines if

Now since we know that S-S' represents two right lines (one finite, the other infinitely distant), it is evident that 1 must be a root of this equation. And it is in fact divisible by k + 1, the quotient being

Ex. 5. Let S represent the parabola y2 - 4mx, and S' the circle as before.

Ans. ▲ 4m2, 0 = — · 4m (a + m), ✪' = ß2 — 4ma — r2, ▲′ = — q2.

372. To find the condition that two conics S and S' should touch each other. When two points, A, B, of the four intersections of two conics coincide, it is plain that the pair of lines AC, BD is identical with the pair AD, BC. In this case, then,

the cubic

Ak3 + Ok2 + O'k + ▲′ = 0,

must have two equal roots. But it can readily be proved that the condition that this should be the case is

(☺☺′ – 9▲▲')* = 4 (0* — 340′) (℗′′ – 3A′O),

or

Θ*Θ" + 18Δ Δ' ΘΘ' – 27 Δ' Δ" - ΑΔΘ" - 4 Δ' Θ* = 0,

which is the required condition that the conics should touch.

It is proved, in works on the theory of equations, that the left-hand member of the equation last written is proportional to the product of the squares of the differences of the roots of the equation in k; and that when it is positive the roots of the equation in k are all real, but that when it is negative two of these roots are imaginary. In the latter case (see Art. 282), S and S' intersect in two real and two imaginary points: in the former case, they intersect either in four real or four imaginary points. These last two cases have not been distinguished by any simple criterion.

If three points A, B, C coincide the conics osculate and in this case the three pairs of right lines are all identical so that the cubic must be a perfect cube; the condition for this are 3A Θ Θ'

The conditions for double contact are of a

different kind and will be got further on.

Ex. 1. To find by this method the condition that two circles shall touch. Forming the condition that the reduced equation (Ex. 3, Art. 371), p2 + (p2+ ‚‚22— D2)k+r'2 k2=0, should have equal roots, we get r2 + p12 — D2 = ± 2rr' ; D = r±r′ as is geometrically evident.

Ex. 2. The conditions for contact between two conics can be shortly found in the cases of trinomial equations by identifying the equations of tangents at any point given Arts. 127, 130, and are for

fyz + gzx + hxy = 0, √(lx) + J(my) + √(nz) = 0, (fl)3 + (gm)} + (hn)3 = 0,

for

ax2 + by2 + cz2 = 0, fyz + gzx + hxy = 0, (aƒ?)} + (bg2)} + (ch2)} = 0.

Ex. 3. Find the locus of the centre of a circle of constant radius touching a given conic. We have only to write for A, A', 0, ' in the equation of this article, the values Ex. 4 and 5, Art. 371; and to consider a, ẞ as the running coordinates. The locus is in general a curve of the eighth degree, but reduces to the sixth in the case of the parabola. This curve is the same which we should find by measuring from the curve on each normal, a constant length, equal to r. It is sometimes called the curve parallel to the given conic. Its evolute is the same as that of the conic.

The following are the equations of the parallel curves given at full length, which may also be regarded as equations giving the length of the normal distances from any point to the curve. The parallel to the parabola is

p.6 − (3y2 + x2 + 8mx − 8m2) r1 + {3y* + y2 (2x2 - 2mx + 20m2)

+ 8mx3 + 8m2x2 - 32m3x + 16m1} r2 — (y2 — 4mx)2 {y2 + (x − m)2} = 0.