Ex. 7. The centre of the circle circumscribing every triangle, self-conjugate with regard to a parabola, lies on the directrix. This and the next example follow from = 0, (Ex. 5, Art. 371).

Ex. 8. The intersection of perpendiculars of any triangle circumscribing a parabola, lies on the directrix.

Ex. 9. Given the radius of the circle inscribed in a self-conjugate triangle, the locus of centre is a parabola of equal parameter with the given one.

376. If two conics be taken arbitrarily it is in general not possible to inscribe a triangle in one which shall be circumscribed about the other; but an infinity of such triangles can be drawn if the coefficients of the conics be connected by a certain relation which we proceed to determine. Let us suppose that such a triangle can be described; and let us take it for triangle of reference; then the equations of the two conics must be reducible to the form

S = x2 + y2 + z2 - 2yz - 2zx — 2xy = 0,

S' = 2fyz +2gzx + 2hxy = 0.

Forming then the invariants, we have

▲=-4, 0=4 (ƒ+g+h), o' = − (f+g+h)', ▲'=2fgh; values which are evidently connected by the relation *= 440'.* This is an equation of the kind (Art. 371) which is unaffected by any change of axes; therefore, no matter what the form in which the equations of the conics have been originally given, this relation between their coefficients must exist, if they are capable of being transformed to the forms here given. Conversely, it is easy to show, as in Ex. 1, Art. 375, that when the relation holds = 4A0', then if we take any triangle circum

* This condition was first given by Mr. Cayley (Philosophical Magazine, Vol. VI., p. 99) who derived it from the theory of elliptic functions. He also proved, in the same way, that if the square root of k3▲ + k2✪ + k✪' + A', when expanded in powers of k, be A + Bk + Ck2 + &c., then the conditions that it should be possible to have a polygon of n sides inscribed in U and circumscribing V, are for n =

3, 5, 7, &c.

scribing S, and two of whose vertices rest on S', the third must do so likewise.

Ex. 1. Find the condition that two circles may be such that a triangle can be inscribed in one and circumscribed about the other. Let D2 2 — p12 — G; then the condition is (see Ex. 3, Art. 371)

(G − p2)2 + 4p2 (G — p′2) = 0, or (G + r2)2 = 4p2p/2;

whence D2 = r22± 2rr', Euler's well known expression for the distance between the centre of the circumscribing circle and that of one of the circles which touch the three sides.

Ex. 2. Find the locus of the centre of a circle of given radius, circumscribing a triangle circumscribing a conic, or inscribed in an inscribed triangle. The loci are curves of the fourth degree except that of the centre of the circumscribing circle in the case of the parabola, which is a circle whose centre is the focus, as is otherwise evident.

Ex. 3. Find the condition that a triangle may be inscribed in S' whose sides touch respectively S+ IS', S+mS', S+nS'. Let

S = x2 + y2+ 22

· 2 (1 + lƒ) yz − 2 (1 + mg) zx - 2 (1+nh) xỵ, S' = 2fyz+2gzx + 2hxy;

then it is evident that S + IS' is touched by x, &c. We have then
A=- (2 + lf + mg + nh)2 — 2lmnfgh,

→ = 2 (f + g + h) (2 + lf + mg + nh) + 2ƒgh (mn + nl + lm),
0' = − (f + g + h)2 − 2 (l + m + n) fƒgh, ▲′ = 2fgh.

Whence obviously

{0 − ▲′ (mn + nl + lm)}2 = 4 (A + ImnA') {0' + A′ (1 + m + n)}, which is the required condition.

377. To find the condition that the line λx+py + vz should pass through one of the four points common to S and S'. This is, in other words, to find the tangential equation of these four points. Now we get the tangential equation of any conic of the system S+kS' by writing a+ka', &c. for a, &c. in the tangential equation of S, or

+2(gh― af) μv + 2 (hf— bg) vλ + 2 (fg − ch) λμ = 0.

We get thus +kÞ+k2Σ' = 0, where

2

Þ= (bc' + b'c — 2ff') λ2 + (ca' + c'a − 2gg') μ3

+(ab' + a'b — 2hh') v2 + 2 (gh' + g'h — aƒ' — a'ƒ) μv

+ 2 (hf' + h'f — bg' — b'g) vλ + 2 (fg' +ƒ'g— ch' — c'h)λμ.

The tangential equation of the envelope of this system is therefore (Art. 298) p=422'. But since S+kS', and the corresponding tangential equation, belong to a system of conics

passing through four fixed points, the envelope of the system is nothing but these four points, and the equation p2 = 4ΣΣ′ is the required condition that the line λx + μy + vz should pass through one of the four points. The matter may be also stated thus: Through four points there can in general be described two conics to touch a given line (Art. 345, Ex. 4); but if the given line pass through one of the four points, both conics coincide in one whose point of contact is that point. Now p2=42' is the condition that the two conics of the system S+kS', which can be drawn to touch Xx + μy + vz, shall coincide.

It will be observed that =0 is the condition obtained (Art. 335), that the line Xx + y + vz, shall be cut harmonically by the two conics.

378. To find the equation of the four common tangents to two conics. This is the reciprocal of the problem of the last Article, and is treated in the same way. Let Σ and ' be the tangential equations of two conics, then (Art. 298) Σ+ kΣ' represents tangentially a conic touched by the four tangents common to the two given conics. Forming then, by Art. 285, the trilinear equation corresponding to Σ + kΣ' = 0, we get

where

AS+F+A'S' = 0,

F = (BC' + B'C – 2FF') x2 + (CA' + C'A − 2 G G') y2

+ (AB' + A'B− 2HH') z2 + 2(GH'+G'H− AF'— A'F) yz + 2 (HF" + H'F− BG' — B' G) zx + 2 (FG' + F' G – CH' — C'H) xy,

the letters A, B, &c. having the same meaning as in Art. 151. But AS+kF+A'S' denotes a system of conics whose envelope is F2=4AA'SS'; and the envelope of the system evidently is the four common tangents.

The equation F2 = 4AA'SS', by its form denotes a locus touching S and S', the curve F passing through the points of contact. Hence, the eight points of contact of two conics with their common tangents, lie on another conic F. Reciprocally, the eight tangents at the points of intersection of two conics envelope another conic P.

It will be observed that F0 is the equation found, Art. 334,

of the locus of points, whence tangents to the two conics form

a harmonic pencil.*

If S' reduces to a pair of right lines, F represents the pair of tangents to S from their intersection.

Ex. Find the equation of the four common tangents to the pair of conics

ax2 + by2 + cz2 = 0, a'x2 + b'y2 + c′z2 = 0.

Here A = bc, B = ca, Cab, whence

Faa' (bc'+b'c) x2 + bb' (ca' + c'a) y2 + cc' (ab' + a'b) z2,

and the required equation is

{aa' (b'c + b'c) x2 + bb' (ca' + c'a) y2 + cc′ (ab' + a'b) z2}2 ́

=

4abca'b'c' (ax2+ by2 + cz2) (a'x2 + b'y2 + c'z2).

379. The former part of this Chapter has sufficiently shown what is meant by invariants, and the last Article will serve to illustrate the meaning of the word covariant. Invariants and covariants agree in this, that the geometric meaning of both is independent of the axes to which the questions are referred; but invariants are functions of the coefficients only, while covariants contain the variables as well. If we are given a curve, or system of curves, and have learned to derive from their general equations the equation of some locus, U=0, whose relation to the given curves is independent of the axes to which the equations are referred, U is said to be a covariant of the given system. Now if we desire to have the equation of this locus referred to any new axes, we shall evidently arrive at the same result, whether we transform to the new axes the equation U=0, or whether we transform to the new axes the equations of the given curves themselves, and from the transformed equations derive the equation of the locus by the same rule that U was originally formed. Thus, if we transform the equations of two conics to a new triangle of reference, by writing instead of x, y, z,

lx+my+nz, l'x+m'y+n'z, l'x+m"y+n"z;

and if we make the same substitution in the equation F2=4^^'SS', we can foresee that the result of this last substitution can only differ by a constant multiplier from the equation F2=4AA'SS', formed with the new coefficients of S and S. For either form

* I believe I was the first to direct attention to the importance of this conie în the theory of two conics.

represents the four common tangents. On this property is founded the analytical definition of covariants. "A derived function formed by any rule from one or more given functions is said to be a covariant, if when the variables in all are transformed by the same linear substitutions, the result obtained by transforming the derived differs only by a constant multiplier from that obtained by transforming the original equations and then forming the corresponding derived."

380. There is another case in which it is possible to predict the result of a transformation by linear substitution. If we have learned how to form the condition that the line Ax+y+ vz should touch a curve, or more generally that it should hold to a curve, or system of curves, any relation independent of the axes to which the equations are referred, then it is evident that when the equations are transformed to any new co-ordinates, the corresponding condition can be formed by the same rule from the transformed equations. But it might also have been obtained by direct transformation from the condition first obtained. Suppose that by transformation λx + μy + vz becomes

μ

λ (lx+my+nz) + μ (l'x + m'y + n'z) + v (l'x + m'y+n"z), and that we write this X'x+p'y+v'z, we have

x' = 1λ + l'μ + l'v, μ' = mλ + m'μ +m"v, v' = nλ+n'μ+ n′′v. Solving these equations, we get equations of the form λ=Lλ'+ L'μ'+L"v', μ=MX'+M'μ'+M"v, v=Nλ'+N'μ'+N'v'. If then we put these values into the condition as first obtained in terms of λ, μ, v, we get the condition in terms of λ', μ', v', which can only differ by a constant multiplier from the condition as obtained by the other method. Functions of the class here considered are called contravariants. Contravariants are like covariants in this: that any contravariant equation, as for example, the tangential equation of a conic, (bc-f") x2 + &c. = 0 can be transformed by linear substitution into the equation of like form (b'c'-ƒ") λ"+ &c. = 0, formed with the coefficients of the transformed trilinear equation of the conic. But they differ in that λ, μ, v are not transformed by the same rule as x, y, z; that is, by writing for A, lλ+ mμ+nv, &c., but by the different rule explained above.

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