the letters A, B, &c. having the same meaning as in Art. 151. But AS+kF+kA'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 F'=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 . It will be observed that F=0 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 common tangents to the pair of conics ax2+by+cz2 = 0, ⋅ a'x2 + b'y2 + c'z2 = 0. Here A = bc, B = ca, C= ab, whence F = aa' (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), which is easily resolved into the four factors x J{aa' (bc')} ±y √{bb' (ca')} ± z √{cc′ (ab′)} = 0. 378a. If S and S' touch, F touches each at their point of contact. This follows immediately from the fact that F passes through the points of contact of common tangents to S and S'. Similarly if S and S' touch in two distinct points, F also has double contact with them in these points. This may be verified by forming the F of cz2+2hxy, c'z2+2h'xy which is found to be of the same form, viz. 2cc'hh'z2 + 2hh′ (ch' + c'h) xy. From what has been just observed, that when S and S' have double contact, F is of the form IS+mS', we can obtain a system of conditions that two conics may have double contact. For write the general value of F, given Art. 334, ax2 + by* + cz2 + 2fyz + 2gzx + 2hxy, * I believe I was the first to direct attention to the importance of this conic in the theory of two conics. Y Y. That when S and S' connected by a linear с f f, g, h = = 0. have double contact, S, F and S' are relation, may be otherwise seen, as follows: When S and S' have double contact there is a value of k for which kS+S' represents two coincident right lines. Now the reciprocal of a conic representing two coincident right lines vanishes identically. Hence we have ΕΣ + ΕΦ + Σ' = 0 identically. But the value of k, for which this is the case, is the double root of the equation k3A+k2 + k© ́ + A′ = 0. Eliminating k between the former equation and the two differentials of the latter we have Σ, Σ', Þ satisfying the identical relation Σ, Φ, Σ' 3A, 20, O' O, 20', 3A'=0. When two conics have double contact their reciprocals have double contact also; and it may be seen without difficulty that the relation just written between 2, ', o implies the following between S, S', F S, F, S 3Δ, 2ΔΘ', Θ Θ', 2 Δ' Θ, 3 Δ' = 0. 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 F-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 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 λx+uy + 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 coordinates, 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 λ (lx+my+nz) + μ (l'x + m'y + n'z) + v (l'x + m'y+n"z), and that we write this 'x+u'y + v'z, we have x' = lλ + 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=LX'+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 X', u', 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 contra variant equation, as for example, the tangential equation of a conic (bc —ƒ2) λ2 + &c. = 0 can be transformed by linear substitution into the equation of like form (b'c' -ƒ")\"2 +&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 λ, la+mμ + nv, &c., but by the different rule explained above. The condition =0 found, Art. 377, is evidently a contravariant of the system of conics S, S'. 381. It will be found that the equation of any conic covariant with S and S' can be expressed in terms of S, S' and F; while its tangential equation can be expressed in terms of 2, 2', Þ. Ex. 1. To express in terms of S, S', F the equation of the polar conic of S with respect to S. From the nature of covariants and invariants, any relation found connecting these quantities, when the equations are referred to any axes, must remain true when the equations are transformed. We may therefore refer S and S' to their common self-conjugate triangle and write Sax2 + by2 + cz2, S' = x2 + y2 + z2. It will be found then that F = a (b + c) x2 + b (c + a) y2 + c (a + b) 22. Now since the condition that a line should touch S is bcλ2+ caμ2 + abv2 = 0, the locus of the poles with respect to S' of the tangents to S is bcx2 + cay2 + abz2 = 0. But this may be written (bc+ca + ab) (x2 + y2 + 22) = F. The locus is therefore (Ex. 1, Art. 371) OS' = F. In like manner the polar conic of S' with regard to S is O'S = F. Ex. 2. To express in terms of S, S′, F the conic enveloped by a line cut harmonically by S and S'. The tangential equation of this conic (b + c) λ2 + (c + a) μ2 + (a + b) v2 = 0. Hence its trilinear equation is or or or = 0 is (c + a) (a + b) x2 + (a + b) (b + c) y2 + (c + a) (b + c) z2 = 0, Ex. 3. To find the condition that F should break up into two right lines. It is abc (b+c) (c + a) (a + b) = 0, or abc {(a + b + c) (bc + ca + ab) — abc} = 0, = which is the required formula. 00' AA' is also the condition that should break up into factors. This condition will be found to be satisfied in the case of two circles which cut at right angles, in which case any line through either centre is cut harmonically by the circles, and the locus of points whence tangents form a harmonic pencil also reduces to two right lines. The locus and envelope will reduce similarly if D2 = 2 (p2 + p′2). V Ex. 4. To reduce the equations of two conics to the forms x2 + y2+ z2 = 0, ax2 + by2 + cz2 = 0. The constants a, b, c are determined at once (Ex. 1, Art. 371) as the roots of A3 - Ꮎk2 + Ꮎk - A' = 0. And if we then solve the equations x2 + y2 + z2 = S, ax2 + by2 + cz2 = S', a (b + c) x2 + b (c + a) y2 + c (a + b) z2 = F, we find x2, y2, z2 in terms of the known functions S, S', F. Strictly speaking, we ought to commence by dividing the two given equations by the cube root of A, since we want to reduce them to a form in which the discriminant of S shall be 1. But it will be seen that it will come to the same thing if leaving S and S'unchanged, we calculate F from the given coefficients and divide the result by A. Ex. 5. Reduce to the above form 3x2-6xy+9y2 - 2x + 4y = 0, 5x2 - 14xy + 8y2. 6x-2=0. It is convenient to begin by forming the coefficients of the tangential equations A, B, &c. These are -4, -1, 18; −3, 3, −2; -16, 19, -9; 21, 24, – 14. We have then ▲ = −9, 0=-54, '99, A'= - 54, whence a, b, c are 1, 2, 3. We next calculate F which is Ex. 6. To find the equation of the four tangents to S at its intersections with S'. Ans. (OS-AS')2 = 4AS (O'S - F). - Ex. 7. A triangle is circumscribed to a given conic; two of its vertices move on fixed right lines λx + μy + vz, λ'x + μ'y + v'z; to find the locus of the third. It was proved (Ex. 2, Art. 272) that when the conic is 22 - xy, and the lines ax y, bx-y, the locus is (a + b)2 (z2 — xy) = (a - b)2 z2. Now the right-hand side is the square of the polar with regard to S of the intersection of the lines, which in general would be P = (ax+hy+gz) (uv′ — μ'v) + (hx+by+ƒz) (v\' − v'λ) + (gx+fy + cz) (λμ' — λ'μ) = 0, and a + b = 0 is the condition that the lines should be conjugate with respect to S, which in general (Art. 373) is → = 0, where → = Aλλ' + Bμμ' + Cvv' + F' (uv' + μ'v) + G (vλ' + v′λ) + H (λμ' + X'μ) = 0. The particular equation, found Art. 272, must therefore be replaced in general by O2U + AP2 = 0. Ex. 8. To find the envelope of the base of a triangle inscribed in S and two of whose sides touch S'. Take the sides of the triangle in any position for lines of reference, and let S = 2 (fyz + gzx + hxy), S' = x2 + y2 + z2 — 2yz — 2zx — 2xy - 2hkxy, where x and y are the lines touched by S'. Then it is obvious that kS+ S' will be |