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 U ̧x + U2y + U ̧z=0, V1x+ V ̧y+ V ̧≈ = 0, W1x + W1y + W ̧z=0, and we obtain the determinant U1 (V ̧W ̧– V ̧W ̧)+ U ̧(V ̧W,− V ̧W ̧) + U ̧ (V ̧W ̧– V ̧W1) = 0. 2 8 2 2 3 1 1 2 2 It is evident that when the polars of any point with respect to U, V, W meet in a point, the polar with respect to all conics of the system lU+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+m V + n W represent two lines, their intersection lies on the Jacobian. For (Art. 292) it satisfies the three equations lU1 + mV, + n W1 =0, lỤ ̧+m V ̧+nW ̧=0, 1U3+mV ̧+n W ̧= 0; 1 2 2 2 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 IU+ mV + nW. The general equation of the Jacobian is (ag'h") x + (bh'f'') y3 + (cf′′g′′) z3 − {(ab′g′′)+ (ah'ƒ'')}x2y—{(ca′h")+(af'g'′′)}x*z—{(ab′ƒ'')+ (bg′h")}y".x -{(bc'h") + (bf'g')}y3z—{(ca′ƒ'')+(cg′h')}z2x—{(bc′g′′) + (ch'f'')} z'y - {(ab'c')+2(fg'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 λU + μV 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'y2 + c'z2, a′′x2 + b′′y2 + c'z2 is xyz = 0. 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. 388 (a). We return now to the theory of two conics which it was not possible to complete until we had explained the nature of Jacobians. We have seen that a system of two conics S, S' has four invariants A, O, O', A', and a covariant conic F, but there is besides a cubic covariant. In fact, the covariant conic F has a common self-conjugate triangle with S, S' (Art. 381, Ex. 1), therefore (Art. 388, Ex. 2) if we form J the Jacobian of S, S', F we obtain a cubic covariant, which, in fact, represents the sides of the common self-conjugate triangle of S end S'. It appears from (Art. 378a) that J vanishes identically if S and S' have double contact. We have given (Art. 381, Ex. 4) another method of obtaining the equation of the sides A A A. of the common self-conjugate triangle, and if we compare the results of the two methods, we get the identical equation J2 = F3 – F2 (©S′ + O ́S) + F (A ́©S2 + AO'S′′) +FSS' ('- 3AA') - A"AS" - A'A'S" + A′ (24℗′ – Ø3) S2 S′ + ▲ (2A′O – O'2) SS””. Thus we see that a system of two conics has, besides the four invariants, four covariant forms S, S', F, J, these being connected by the relation just written. In like manner, there are four contravariant forms Σ, ', 4, г, where the last expresses tangentially the three vertices of the self-conjugate triangle, its square being connected by a relation, corresponding to that just written, between 2, ', o and the invariants. Ex. 1. Write down the 12 forms for the conics x2 + y2 + z2, ax2 + by2 + cz2. S = x2 + y2+22, S'= ax2 + by2+ cz2, F=a (b+c) x2+b(c+a) y2+c (a+b) z2, Σ =λ2 + μ2 + v2, Σ' = bcλ2 + caμ2+abv2, §= (b+c) λ2+(c+a) μ2+(a+b) v2, Ex. 2. Find an expression for the area of the common conjugate triangle of two conics. The square of the area is found to be where M is the area of the triangle of reference, and I' the result of substituting in г, sin A, sin B, sin C, the coordinates of the line at infinity. That the expression must contain in the numerator the condition of contact, and in the denominator I', is evident from the consideration that this area must vanish if the conics touch, and becomes infinite if any vertex of the triangle be at infinity. 388 (6). We have already explained what is meant by covariants which express relations satisfied by x, y, z, the coordinates of a point lying on a locus having some permanent relation with the original curve or curves, and by contravariants which express relations satisfied by λ, u, v the tangential coordinates of a line, whose section by the original curve or curves has some property unaffected by transformation of coordinates. There are besides forms called mixed concomitants which contain both x, y, z and also λ, u, v, and these we proceed These to enumerate for the system of two conics S, S'. mixed concomitants of a system of two curves may also be regarded as covariants of the system of three, consisting of S, S' and the right line λx+μy + vz. For instance, we may form the Jacobian of that system, or the locus of the point whose polars, with respect to S and S', intersect on λx + μy + vz, thus obtaining the mixed concomitant N or λ μ ν which for the canonical form is S1, S2, S. S', S, S' λ (b − c) yz + μ (c − a) zx + v (a − b) xy. There is evidently a corresponding reciprocal form N' obtained in the same way from Σ, ', which for the canonical form is aμv (b −c) x+bvλ (c − a) y + cλμ (a – b) z. This expresses the equation of the line joining the poles of λx+μy+vz with respect to S and S'. Again, for any line λx +μy + vz, we may take its pole with regard to S and again the polar of that point with regard to S' and so obtain a companion line K. This for the canonical form is We obtain a different companion line K' by taking the pole with regard to S′ and then the polar with regard to S, thus finding bcλx + caμy + abvz. Gordan has shewn (Clebsch, Geometrie, p. 291) that there are in all eight mixed concomitants of a system of two conics in terms of which, and of the forms previously enumerated, all other concomitants can be expressed. In addition to the four already mentioned we may take the Jacobian of K, S and λx+μy+vz, or for the canonical form μv (b −c) x + vλ (c− a) y +λμ (a − b) z ; and, in like manner, the Jacobian of K′, S', and λx + μy + vz, or μva2 (b−c) x + vλb2 (c − a) y +λμc2 (a — b) z. These with the two reciprocal forms and Xbc (b − c) yz + μca (c − a) zx + vab (a – b) xy make up the entire system. We return now to the theory of three conics. 388(c). To find the condition that a line λx+μy+vz should oe cut in involution by three conics. It appears from Art. 335 and from the Note, Art. 342, that the required condition is the vanishing of the determinant ελ 2gλ +av, εμ – 2fνμ +όν, ελμ fνλ -γνμ +hv c'λ2 −2g'vλ +a'v3, c'μ3 — 2ƒ'vμ +b′v3, c'λμ —ƒ ̃vλ —g'vμ +h'v2 c′′X2−2g′′vλ+a′′v3, c′′ μ3— 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 λ3 (bc'f'') + μ3 (ca'g′′) + v3 (ab′h′′) +λ3μ {2 (ch'f'"') — (bcʻg′′)} - +λ3v {2 (bƒ'g′′) − (bc'h′′)} + μ3λ {2 (cg′h") — (ca'ƒ'')} +μ3v {2 (aƒ'g′′) – (ca′h")} + v2λ {2 (bgʻh′′) — (ab'ƒ'')} + v3μ {2 (ah'ƒ'') – (ab'g')} +λμv {(ab'c′′) − 4 (fg'′h′′)} = 0. This may also be written in the determinant form 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 U+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, v in the preceding, and the reciprocal coefficients A, B, &c. instead of a, b, &c. 11 389. If we form the discriminant of IU+ mV + n W, we may write the result 3▲ + l'mo„ + l'n0118 + Imn0123 + &c., and 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'" + al"c' + a'b"c+ abc" + a′′bc' + a′′b'c, &c. Another remarkable invariant of the system of conics, first obtained by a different method by Prof. Sylvester, is found by the help of the principle (Higher Algebra, Art. 139), that when we have a covariant and a contravariant of the same degree, we |