imaginary conic; that all circles on a sphere are to be considered as conics having double contact with a fixed conic, the centre of the circle being the pole of the chord of contact; that two lines are perpendicular if each pass through the pole of the other with respect to that conic, &c. The theorems then, which, in the Chapter on Projection, were extended by substituting, for the two imaginary points at infinity, two points situated anywhere, may be still further extended by substituting for these two points a conic section. Only these extensions are theorems suggested, not proved. Thus the theorem that the intersection of perpendiculars of a triangle inscribed in an equilateral hyperbola is on the curve, suggested the property of conics connected by the relation = 0, proved at the end of Art. 375. It has been proved (Art.306) that to several theorems concerning systems of circles, correspond theorems concerning systems of conics having double contact with a fixed conic. We give now some analytical investigations concerning the latter class of systems. 386. To form the condition that the line λx+μy+vz may touch S+ (x + μ'y + v'z). We are to substitute in Σ, a+λ22, b+μ", &c. for a, b, &c. The result may be written Σ+ {a (μv' — μ'v)2 + &c.} = 0, μ' - λ' μ for For it was where the quantity within the brackets is intended to denote the result of substituting in S μν' - μ'ν, νλ' - ύλη x, y, z. This result may be otherwise written. proved (Art. 294) that (ax2+&c.) (ax* + &c.) − (axx' + &c.)3 = A (yz′ — y′z)2 +&c. And it follows, by parity of reasoning, and can be proved in like manner, that (Ax2+&c.) (Aλ'2 + &c.) — (Anλ'+ &c.)2 = ▲ {a (μv'— μ'v)2+ &c.}, where AXX' + &c. is the condition that the lines λx + μy + vz, x'x + μ'y+v'z may be conjugate; or Aλλ'+ Bμμ'+ Cvv′ + F (μv'+ μ ́v) + G (vX' + vλ) + H(λμ'+X'μ).; If then we denote Aλ"+&c. by ', and Axλ'+&c. by П and if we substitute for a (uv'p'v)2+ &c. the value just found, the condition previously obtained may be written (Δ + Σ') Σ - Π* = 0. If we recollect (Art. 321) that λ, μ, v may be considered as the coordinates of a point on the reciprocal conic, the latter form may be regarded as an analytical proof of the theorem that the reciprocal of two conics which have double contact is a pair of conics also having double contact. This condition may also be put into a form more convenient for some applications, if instead of defining the lines λx + μy + vz, &c. by the coefficients λ, μ, v, &c., we do so by the coordinates of their poles with respect to S, and if we form the condition that the line P' may touch S+ P', where P' is the polar of x'y'z', or axx' + &c. Now the polar of x'y'z' will evidently touch S when x'y'z' is on the curve; and in fact if in Σ we substitute for λ, μ, v; S1, S., S, the coefficients of x, y, z in the equation of the polar, we get ▲S. And again two lines will be conjugate with respect to S, when their poles are conjugate; and in fact if we substitute as before for λ, μ, v in II we get AR, where R denotes the result of substituting the coordinates of either of the points x'y'z', x"y"z", in the equation of the polar of the other. The condition that P' should touch S+ P'" then becomes (1+ S′′) S′ = R2. 387. To find the condition that the two conics S+ (Xx + μ'y + v'z)2, S+ (N′′x+ μ”y + v′′z)3, should touch each other. They will evidently touch if one of the common chords (x+μ'y + v ́z) ± (X′′ x + μ'”y+v"z) touch either conic. Substituting, then, in the condition of the last Article X" for λ, &c., we get (Δ + Σ') (Σ' ± 2Π + Σ") = (Σ' + Π), which reduced may be written in the more symmetrical form (A + Σ' ́) (A + Σ'') = (A ± II)2. The condition that S+P" and S+ P"" may touch is found from this as in the last Article, and is (1 + S′) (1 + S′′) = (1 ± R)2. Ex. 1. To draw a conic having double contact with S and touching three given conics S + P2, S + P''2, S + P'', also having double contact with S. Let xyz be the coordinates of the pole of the chord of contact with S of the sought conic S + P2, then we have (1 + S) (1 + S′) = (1 + P′)2 ; (1 + S) (1 + ́S”) = (1 + P'')2 ; (1 + S) (i + S'''') = (1 + P''')2 where the reader will observe that S', S", S"" are known constants, but S, P', &c. involve the coordinates of the sought point xyz. If then we write 1+ S = k2, &c., we get kk' = 1+ P', kk" =1+ P", kk'"' = 1+ P'"'. It is to be observed that P', P", P"" might each have been written with a double sign, and in taking the square roots a double sign may, of course, be given to k', k", k". It will be found that these varieties of sign indicate that the problem admits of thirty-two solutions. The equations last written give k (k' — k') = P' — P"; k (k" — k"") = P" — P'"' ; whence eliminating k, we get P' (k" — k''') + P" (k”” — k') + P''' (k' — k'') = 0, the equation of a line on which must lie the pole with regard to S of the chord of contact of the sought conic. This equation is evidently satisfied by the point P = P" = P"". But this point is evidently one of the radical centres (see Art. 306) of the conics S+ P'2, S+ P"2, S+ P'"'2. geometric interpretation of this we remark that it may be deduced from Art. 386 that the tangential equations of S + P'2, S + P2 are respectively respect to S, of an axis of similitude (Art. 306) of the three given conics. And the theorem we have obtained is,-the pole of the sought chord of contact lies on one of the lines joining one of the four radical centres to the pole, with regard to S, of one of the four ares of similitude. This is the extension of the theorem at the end of Art. 118. To complete the solution, we seek for the coordinates of the point of contact of S+ P2 with S + P'2. Now the coordinates of the point of contact, which is a centre where R, R' are the results of substituting "y"z", x""y""z"" respectively in the polar the equation of a line on which the sought point of contact must lie; and which evidently joins a radical centre to the point where P', P", P'"' are respectively pro form the equations of the polars, with respect to S+ P2, of the three centres of similitude as above, we get (k'k" — R) P′ = P", (k'k''' — R') P' = P'"', &c., showing that the line we want to construct is got by joining one of the four radical centres to the pole, with respect to S+ P'2, of one of the four axes of similitude. This may also be derived geometrically as in Art. 121, from the theorems proved, Art. 306. The sixteen lines which can be so drawn meet S + P2 in the thirty-two points of contact of the different conics which can be drawn to fulfil the conditions of the problem.* *The solution here given is the same in substance (though somewhat simplified in the details) as that given by Prof. Cayley, Crelle, vol. XXXIX. Prof. Casey (Proceedings of the Royal Irish Academy, 1866) has arrived at another solution from considerations of spherical geometry. He shows by the method used, Art. 121 (a), that the same relation which connects the common tangents of four circles touched by the same fifth connects also the sines of the halves of the common tangents 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-L20, S― M2 = 0, S — N2 = 0, that of a group of circles touching all three will be of the form d{λ (s3 — L)} + ↓{μ (S3 − M)} + √{v (S* − N)} = 0. This evidently gives a solution of the problem in the text, which I have arrived at directly by the following process. Let the conic S be x2 + y2+ z2, and let L = lx + my + nz, M = l'x + m'y + n'z; then the condition that S− L2, S — M2 should touch is (Art. 387) (1 − S′) (1 − S′′) = (1 − K)2, where S′ = 12 + m2 + n2, S"=l'2+m'2+n'2, R=ll'+mm'+nn'. I write now (12) to denote √(1—S′′′)(1—S”)—(1— R). Let us now, according to the rule of multiplication of determinants, form a determinant from the two matrices containing five columns and six rows each. The resulting determinant which must vanish, since there are more rows than columns, is 1, 1, 1, 1, 1 0, = 0, an identical relation connecting the invariants of five conics all having double contact with the same conic S. Suppose now that the conic (5) touches the other four, 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 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) + 2 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, v between Αλ3 – θλέμ + θ'λμ? - Δ' μ* = 0, and the two corresponding equations which express that μV- vW, vW - λU break up into right lines. then (15), &c. vanish; and we learn that the invariants of four conics all having double contact with S and touched by the same fifth are connected by the relation or 0, (12), (13), (14) (12), 0, (23), (24) (13), (23), 0, (34) (14), (24), (34), = 0, J{(12) (34)} ± √{(13) (24)} ± √{(14) (23)} = 0. We may deduce from this equation as follows the equation of the conic touching three others. If the discriminant of a conic vanish, S = 1, and then the condition of contact with any other reduces to R = 1. If, then, a, ß, y be the coordinates of any point satisfying the relation S- L2 = 0, or x2 + y2 + z2 — (lx + my + nz)2 = 0, then evidently denotes a conic whose discriminant vanishes and which touches S-L. If, then, we are given three conics S-L2, S- M2, S - N2, take any point a, ß, y on the conic which touches all three and take for a fourth conic that whose equation has just been written, then the functions (14), (24), (34) are respectively J[(23) {√(S) — L}] ± √[(31) {√(S) — M}] ± √[(12) {√(S) — N}] = 0. *This is an extension of Feuerbach's theorem (p. 127), and itself admits of further extension. See Quarterly Journal of Mathematics, vol. vI. p. 67. |