Ex. 2. The four conics having double contact with a given one S, which can be drawn through three fixed points, are all touched by four other conics also having double contact with S.† Let

S = x2 + y2+22 – 2yz cos A – 2zx cos B-2xy cos C,

then the four conics are S = (x + y + z)2, which are all touched by

S = {x cos (BC) + y cos (C′ − A) + ≈ cos (A — B)}2,

and by the three others got by changing the sign of A, B, or C, in this equation.

Ex. 3. The four conics which touch x, y, z, and have double contact with S are all touched by four other conics having double contact with S. Let M=¿ (A + B + C'), · then the four conics are

S = {x sin (M – A) + y sin (M − B) + z sin (M — C')}2,

together with those obtained by changing the sign of A, B, or C in the above; and one of the touching conics is

the others being got by changing the sign of x, and at the same time increasing B and C by 180°, &c.

Ex. 4. Find the condition that three conics U, V, W shall all have double contact with the same conic.. The condition, as may be easily seen, is got by eliminating λ, μ, ν between

Δλο – θλέμ +θ'λμ? - Δ' με = 0,

and the two corresponding equations which express that μV-W, vW – λU break up into right lines.

388. The theory of invariants and covariants of a system of three conics cannot be fully explained without assuming some knowledge of the theory of curves of the third degree.

Given three conics U, V, W, the locus of a point whose polars with respect to the three meet in a point is a curve of the third degree; which we call the Jacobian of the three conics. For we have to eliminate x, y, z between the equations of the three polars U1x+U2y+ U ̧z=0, V1x+V2y+V ̧2=0, W1x+W2y+W ̧z=0, and we obtain the determinant

1

2

8

3

24

2

3

1

U1 (V ̧W ̧− V ̧W ̧) +U ̧(V ̧W, − V ̧W1)+U ̧(V2W ̧− V2W1) = 0. It is evident that when the polars of any point with respect to

1

gents of four such circles on a sphere; and hence, as in Art. 121 (6), that if the equations of three circles on a sphere (see Geometry of Three Dimensions, Chap. IX.) be S – L2 = 0, S — M2 = 0, S — N2 = 0, that of a group of circles touching all three will be of the form

√{\ (S1 — L)} + √{μ (S1 — M)} + √{v (S3 — N)} = 0.

This evidently gives a solution of the problem in the text, but I have not succeeded in arriving at it directly. The constants A, u, v are, I believe, found by forming for each pair of conics ▲ - II - √{(A — Σ') (4 − Σ'')}.

†This is an extension of Feuerbach's theorem (p. 126) and itself admits of further extension, See Quarterly Journal of Mathematics, Vol. VI., p. 67.

YY

U, V, W meet in a point, the polar with respect to all conics of the system U+mV+nW will pass through the same point. If the polars with respect to all these conics of a point A on the Jacobian pass through a point B, then the line AB is cut harmonically by all the conics; and therefore the polar of B will also pass through A. The point B is, therefore, also on the Jacobian, and is said to correspond to A. The line AB is evidently cut by all the conics in an involution whose foci are the points A, B. Since the foci are the points in which two corresponding points of the involution coincide, it follows that if any conic of the system touch the line AB, it can only be in one of the points A, B; or that if any break up into two right lines intersecting on AB, the points of intersection must be either A or B, unless indeed the line AB be itself one of the two lines. It can be proved directly, that if lU+mV+nW represent two lines, their intersection lies on the Jacobian. For (Art. 292) it satisfies the three equations

1

2

2

IU1+mV1+nW1 =0, lU2+mV2+nW1=0, lỤ2+mV ̧+nW ̧=0; whence, eliminating l, m, n, we get the same locus as before. The line AB joining two corresponding points on the Jacobian meets that curve in a third point; and it follows from what has been said that AB is itself one of the pair of lines passing through that point, and included in the system U+mV+nW. The general equation of the Jacobian is

(ag'h') x3 + (bh'f') y3 + (cf'g′′) ≈3

- {(ab'g")+(ah'f")} x'y- {(ca'h")+(af'g')} x2z — {(ab'f") + (bg'h") y2x - {(bc'h")+(bf'g")}yz - {(ca'f")+(cg'h")} #2x-{(bc'g")+(ch'f")}z2y

— {(ab'c") +2 (ƒg'h")} xyz = 0,

where (ag'h") &c. are abbreviations for determinants.

Ex. 1. Through four points to draw a conic to touch a given conic W. Let the four points be the intersection of two conics U, V; and it is evident that the problem admits of six solutions. For if we substitute a+ka', &c. for a in the condition (Art. 372) that U and W should touch each other, k, as is easily seen, enters into the result in the sixth degree. The Jacobian of U, V, W intersects W in the six points of contact sought. For the polar of the point of contact with regard to W being also its polar with regard to a conic of the form XU+uV passes through the intersection of the polars with regard to U and V.

Ex. 2. If three conics have a common self-conjugate triangle, their Jacobian is three right lines. For it is verified at once that the Jacobian of ax2 + by2 + cz2,

a'x2 + b'y3 + c'z3, a′′x2 + b′′y2 + c'z2 is xyz=0. We can hence find at once the equation of the sides of the common self-conjugate triangle of two conics, by forming the Jacobian of S, S' and the covariant F; since this triangle is also self-conjugate with respect to F (Art. 381, Ex. 1).

Comparing this with the result obtained by Art. 381, Ex. 4, we get the identical equation

J2 = F3 — F2 (OS' + O'S) + F (4′OS2 + AO'S'2) + (00′ – 3AA') FSS'

— A′2AS3 — ▲▲′2S'3 + A′ (2A0′ — O2) S2S' + A (2A′O — O'2) SS”.

Ex. 3. If three conics have two points common, their Jacobian consists of a line and a conic through the two points. It is evident geometrically that any point on the line joining the two points fulfils the conditions of the problem, and the theorem can easily be verified analytically. In particular the Jacobian of a system of three circles is the circle cutting the three at right angles.

Ex. 4. The Jacobian also breaks up into a line and conic if one of the quantities S be a perfect square L2. For then L is a factor in the locus. Hence we can describe four conics touching a given conic S at two given points (S, L) and also touching S"; the intersection of the locus with S" determining the points of contact.

When the three conics are a conic, a circle, and the square of the line at infinity, the Jacobian passes through the feet of the normals which can be drawn to the conic through the centre of the circle.

388a. To find the condition that a line λx+μy+vz should be cut in involution by three conics. It appears from Art. 335 and from the Note, p. 298, that the required condition is the vanishing of the determinant

ελ 2gλ +av, cμ – 2fνμ +bv ελμ - fνλ -9νμ +hv* c'λ2 −2g'vλ +a'v2, c'μ2 −2ƒ'vμ +b'v2, c'λμ −ƒ'vλ − g'vμ +h'v2 c"λ2--2g"vλ+a"v2, c"μ2-2ƒ"vμ+b"v2, c"λμ—ƒ"vλ−g′′vμ+h"v2 When this is expanded, it becomes divisible by v3, and may be written

3

λ3 (bc'f") +μ3 (ca'g") + v2 (ab'h") +λ3μ {2 (ch'f") — (be'g")}

+λ2v {2 (bf'g'') — (bc'h'')} + μ3λ {2 (cg'h') — (ca'ƒ')}

+uv (2 (af'g")-(ca'h")} +v3λ {2 (bg'h") — (ab'f")}

2

+ v3μ {2 (ah'f") — (ab'g")} + λμv {(ab'c") - 4 (fg'h")}=0, From the form of this condition, it is immediately inferred that any line cut in involution by three conics U, V, W is cut in involution by any three conics of the system lU+mV+nW. The locus of a point whence tangents to three conics form a system in involution, is got by writing x, y, z for λ, μ, in the preceding, and the reciprocal coefficients A, B, &c. instead of α, b, &c.

389. If we form the discriminant of lU+mV+nW, the coefficients of the several powers of l, m, n will be invariants of the system of conics. All these belong to the class of invariants already considered, except the coefficient of lmn, in which each term abc of the discriminant of U is replaced by

ab'c' + ab"c' + a'b"c+ a'bc" + a′′bc' +a"b'c, &c.

Another remarkable invariant of the system of conics, first obtained by a different method by Mr. Sylvester, is found by the help of the principle (Higher Algebra, p. 110), that when we have a covariant and a contravariant of the same degree, we can get an invariant by substituting differential symbols in either, and operating on the other. By the help of the Jacobian and the contravariant of the last article we get the invariant T, T'= (ab'c')2 + 4 (ab'f") (ac'f") + 4 (bc'g") (ba'g'") + 4 (ca'h") (cb'h") +8 (af'g") (bf'g") +8 (af'h") (cf'h") +8 (cg'h") bg'h") — 8 (ag'h") (bc'ƒ") — 8 (bh'f") (ca'g") — 8 (cf'g′′) (ab'h") + 4 (ab'c") (fg'h') — 8 (fg'h")2.

XV

389a. Some of the properties of a system of three conics can be studied with advantage by expressing each in terms of four lines x, y, z, w: thus

U=ax2+by+cz2 + dw*, V = a'x2 + b'y2+ c'z2 + d'w2,

W = a"x2+b"y" +c"z2+d"w2.

It is always possible, in an infinity of ways, to choose x, y, z, w so that the equations can be brought to the above form: for each of the equations just written contains explicitly three independent constants: and each of the lines x, y, z, w contains implicitly two independent constants. The form, therefore, just written puts seventeen constants at our disposal, while U, V, W, contain only three times five, or fifteen, independent constants. The equations of four lines are always connected by a relation of the form wλx + μy+vz, and we may suppose that the constants λ, &c. have been included in x, &c., so that this relation may be written in the symmetrical form a+y+z+w=0.

Let it be required now to find the condition that U, V, W may have a common point. Solving for x, y, z2, w2 between the equations U=0, V=0, W=0, and denoting by A, B, C, D

the four determinants (bc'd"), (dc'a"), (da'b"), (ba'c"), we get x2, y2, z2, w2 proportional to A, B, C, D; and substituting in x+y+z+w=0, we obtain the required condition

√ (A) + √ (B) + √ (C) + √ (D) = 0,

or (A2+B2+C+D2− 2AB−2BC-2 CA-2AD-2BD – 2CD)2 =64ABCD.

The left-hand side of this equation is the square of the invariant T already found; the right-hand side ABCD is an invariant which we shall call M, whose vanishing expresses the condition that it may be possible to determine l, m, n, so that lU+mV+nW shall be a perfect square. Since M is of the fourth degree in the coefficients of each conic, it follows that four conics of the system S+1U+mV+n W can be determined so as to be perfect squares (see Ex. 3, p. 327), for if we equate to nothing the invariant M found for S+1U, V, W, we have an equation of the fourth degree for determining 7.

3896. Any three conics may in general be considered as the polar conics of three points with regard to the same cubic; or, in other words, their equations may all be reduced to the form

a (x2 – 2yz) +ẞ (y2 — 2zx) + y (≈2 — 2xy) = 0.

If we use for the equations of the conics the forms given in the last article, the equation of the cubic whence they are derived will be

and it appears that if the invariant M vanish (in which case either A, B, C or D vanishes), an exception occurs, and the conics cannot all be derived from the same cubic. In the general case the equation of the cubic may be obtained by forming the Hessian of the Jacobian of the three conics, and subtracting the Jacobian itself multiplied by T.

If we operate with the conics on the cubic contravariant, or with their reciprocals on the Jacobian, we obtain linear contravariants and covariants which geometrically represent the points of which the given conics are polar conics, and the polar lines of these points with respect to the cubic.