opposite vertices of the quadrilateral formed by the common tangents. In the case where denotes the circular points at infinity, when + k' represents a pair of points, these points are the foci (Art. 279). If then it be required to find the foci of a conic, given by a numerical equation in Cartesian co-ordinates, we first determine k from the quadratic (ab − h2) k2 + ▲ (a + b) k + ▲2 = 0. - Then, substituting either value of k in Σ + k (λ2 + μ3), it breaks up into factors (λx' + μy' + vz') (λx" + μy" + vz"); and the foci x'y' x" y" y'. Z" " z" are - One value of k gives the two real foci, and the other two imaginary foci. The same process is applicable to trilinear co-ordinates. In general, Σ+k (λ2+μ3) represents tangentially a conic confocal with the given one. Forming, by Art. 285, the corresponding Cartesian equation, we find that the general equation of a conic confocal with the given one is ▲S+k{C (x2 + y2) − 2 Gx − 2 Fy + A + B} + k2 = 0. From this we can deduce that the equation of common tangents is By resolving this into a pair of factors {(x − a)2 + (y − B)2} {(x − a')2 + ( y − B′)2}, we can also get a, ẞ; a', B' the co-ordinates of the foci. -- Ex. 1. Find the foci of 2x2-2xy + 2y2 — 2x − 8y +11. The quadratic here is 3k2 + 4k▲ + A20, whose roots are k value = 3, A, k - A. But A = 9. Using the 6x2+21μ2 + 3v2 + 18μv + 12vλ + 30λμ + 3 (λ2 + μ2) = 3 (λ + 2μ + v) (3λ + μ + v), showing that the foci are 1, 2; 3, 1. 2 ± √(− 1), 3 F √(− 1). The value 9 gives the imaginary foci Ex. 2. Find the co-ordinates of the focus of a parabola given by a Cartesian equation. The quadratic here reduces to a simple equation, and we find that (a + b) {Aλ2 + Bμ2 + 2Fμv + 2Gvλ + 2Hλμ} − ▲ (\2 + μ2) is resolvable into factors. But these evidently must be The first factor gives the infinitely distant focus, and shows that the axis of the curve is parallel to Fx - Gy. The second factor shows that the co-ordinates of the focus are the coefficients of X and μ in that factor. Ex. 3. Find the co-ordinates of the focus of a parabola given by the trilinear equation. The equation which represents the pair of foci is O'Σ = ▲ (\2 + μ2 + v2 – 2μv cos A - 2vλ cos B - 2λμ cos C). But the co-ordinates of the infinitely distant focus are known, from Art. 293, since it is the pole of the line at infinity. Hence those of the finite focus are A sin A+ H sin B + G sinC' H sin A + B sin B + F sin C' Ꮎ Ꮯ - A G sin A+ F sin B + C sin C' 385. The condition (Art. 61) that two lines should be mutually perpendicular λλ' + μμ' + vv' − (μv' + μ'v) cos A − (vλ' + v'λ) cos B is easily seen to be the same as the condition (Art. 293) that the lines should be conjugate with respect to 2 x2 + μ2 + v2 − 2 μv cos A – 2vλ cos B− 2λμ cos C=0. The relation, then, between two mutually perpendicular lines is a particular case of the relation between two lines conjugate with regard to a fixed conic. Thus, the theorem that the three perpendiculars of a triangle meet in a point, is a particular case of the theorem that the lines meet in a point which join the corresponding vertices of two triangles conjugate with respect to a fixed conic, &c. It is proved (Geometry of Three Dimensions, Chap. IX.) that, in spherical geometry, the two imaginary circular points at infinity are replaced by a fixed 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+py+vz may touch S+ (N'x+py + v'z)". We are to substitute in Σ, a+λ'2, b+μ", &c. for a, b, &c. The result may be written 12 Σ+ {a (μv' — μ'v)2 + &c.} = 0, λμ' - 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' - ν'λ, 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 (Aλ2+&c.) (Aλ" + &c.) − (Ann'+ &c.)2 = ▲ {a (μv^ — μ'v)2+&c.}, where A+ &c. is the condition that the lines λx+y+vz, x'x + μ'y + v'z may be conjugate; or Axλ'+ Bμμ' + Cvv' + F (uv' + μ'v) + G (vλ' + v'λ) + H (λμ'+λ'μ). If then we denote Ax" + &c. by E', and Axx'+ &c. by II; and if we substitute for a (uv' — 'v)" + &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 co-ordinates 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 line λx+μy+vz by the coefficients λ, u, v, we do so by the co-ordinates of its pole 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 A, μ, v; S1, S2, S, the coefficients of x, y, z in the equation of the polar, we get AS'. 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 co-ordinates 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 8+ (N'x + μ'y + v'z)", S+ ("x+μ"y+v"z)", should touch each other. They will evidently touch if one of the common chords, ('x + p'y + v'z) ± (N"x + μ"y + v"z), touch either conic. Substituting, then, in the condition of the last Article '+" for λ, &c., we get (A + Σ') (Σ' ± 211 + Σ′′) = (Σ' ± 11)2, which reduced may be written in the more symmetrical form (A + X') (A + Σ'') = (▲ ± 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)3. Ex. 1. To draw a conic having double contact with S and touching three given conics S+ P2, S + P"2, S + P'"2, also having double contact with S. Let xyz be the co-ordinates 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) (1 + S'''') = (1 + P''')2; where the reader will observe that S', S", S"" are known constants, but S, P', &c. involve the co-ordinates 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 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 p. 270) 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 + P22, S + P"2 are respectively represent points of intersection of common tangents to S+ P'2, S + P", that is to say, the co-ordinates of these points are, &c., and the polars of these points, P' p" with respect to S, are ± k' k" respect to S, of an axis of similitude (p. 270) 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 axes of similitude. This is the extension of the theorem at the end of Art. 118. P' P" It follows that = = denote the pole, with k-k' k" To complete the solution, we seek for the co-ordinates of the point of contact of S+ P2 with S + P'2. Now the co-ordinates of the point of contact, which is a centre of similitude of the two conics, being х x' , &c., we must substitute x + x, &c. in the equations kk' =1+ P', &c., and we get k x' for 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+ P2, of one of the four axes of similitude. This may also be derived geometrically as in Art. 121, from the theorems proved, p. 271. The sixteen lines which can be so drawn, meet S+ P'2 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 Mr. Cayley, Crelle, Vol. XXXIX. Mr. Casey (Proceedings of the Royal Irish Academy, 1866) has arrived at another solution from considerations of spherical geometry. He shows by the method used, p. 113, 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 tan |