F is two real straight lines forming a harmonic pencil with the two common chords, and the two points into which F' degenerates are the poles of the two common chords.]

1551. An hyperbola is described whose asymptotes are conjugate diameters of a given ellipse: prove that the relation ′ = AA' is satisfied for the two conics: that when there are four real common points the two points F' are two real points at co, the poles of the common diameters; and the two straight lines F are two impossible diameters: when there are four real common tangents, the points of contact lie on two diameters (the straight lines F) and the points F" are impossible.

1552. The general equation of a conic, for which the relation O' AA' is satisfied with the given conic le+ my2 + nz2 = 0, is

1553. Prove that the equations of two conics, satisfying the relation ☺☺′ = ▲▲', may be always reduced to the forms

and reduce in this manner the two pairs of circles

[(1) (5x-49)-24y+49 (x-5)=0, (5x-49) + 24y2 = (x-5)* ; (2) {(2+i) x-4 (1 + 2i)}2 + {(2 −i) x − 4 (1-2)} + by2 = 0,

{(2 + i) x − 4 (1 + 2i)} - {(2 − i) x − 4 (1 - 2i)}'+ 8iy2 = 0.]

1554. The equations of two conics, for which the relation ′ = ▲▲′ is satisfied, can always be put in the forms (areal co-ordinates)

U = lx2 + my2 + nz2 = 0, U' = x2 + 2pyz=0,

and, if the two straight lines F meet the two curves in P, Q, P', Q'; P, I, P', q'; the ranges {PP'Q'Q}, {pp'qq} will be equal; and similarly the tangents drawn to the two curves from the two points F" will form pencils of equal ratios.

1555. If ABC be the triangle of reference in the last question, the quadrangle formed by the points of contact of the common tangents with either curve will have the same vertices as the quadrangle formed by the points in which AB, AC meet U.

1556. The tangent and normal to a rectangular hyperbola at P meet the transverse axis in T, G, and a circle is drawn with centre G and radius GP: prove that straight lines drawn through T parallel to the asymptotes will pass through the points of contact of common tangents

drawn to this circle and to the auxiliary circle, and the tangents drawn to the two circles from any point on either of these straight lines will form a harmonic pencil.

1557. The harmonic locus and envelope of the conics

where r, r' are the arithmetic and geometric means between p and q.

1558. The harmonic locus and envelope of the conics

2x2 + 2λxy + y2 = 2ax, 2mx2 + 2μxy + y2 = 2ax,

are respectively

1559.

{l + m − } (x −μ)2 } x2 + (λ + μ) xy + y2 — 2ax = 0,

{l + m + }(\ − μ)2 } x2 + (λ + μ) xy + y2 — 2ax = 0.

Prove that when two conics have contact of the third order

the harmonic locus and envelope coincide.

[2mx2 + y2 = 2ax, 2nx2 + y2 = 2ax, (m + n) x2 + y2 = 2ax.]

1560. Four tangents are drawn to a circle U forming a quadrilateral such that the extremities of two of its diagonals lie on another circle U': prove that if a, a' be the radii, and b the distance between the centres,

ba' or (b2-a') = 2a2 (b2 + a'2).

[In the former case U' is the locus of the points from which tangents drawn to U divide harmonically the diameter of U' drawn from the centre of U, and U is the envelope of chords of·U' divided harmonically by the radical axis and by the diameter of U' which is at right angles to the line of centres.]

1561. Two conics will be such that quadrilaterals can be circumscribed to either with the ends of two diagonals on the other, if

= 20'▲, and " = 20A':

and the curves 22 − y3 = a2, x2 + y2 ± 2 √3ax + 2a2 = 0 are so related.

1562. Prove that if a circle and rectangular hyperbola be described so that each passes through the centre of the other, and a parabola be described with its focus at the centre of the hyperbola and directrix touching the hyperbola at the centre of the circle, the three form a harmonic system, such that, if any two be taken as U and U', the covariants F, F, V, V' all coincide with the third, and thus that an infinite number of triangles can be inscribed in the first whose sides touch the second and which are self-conjugate to the third, whatever be the order in which the three are taken.

1563. A straight line is divided harmonically by two conics and its pole with respect to either lies on the other: prove that the same property is true for every other straight line divided harmonically; that, for the two conics, = 0, = 0; and that the harmonic locus and envelope coincide and form with the two a harmonic system.

1564. The conics U,, U, U, form a harmonic system and any triangle ABC is inscribed in U, whose sides touch U, in a, b, c; then abe will be a triangle whose sides touch U, (in A', B′, C') and A'B'C' will be a triangle whose sides touch U, in A, B, C.

3

1565. Prove that any two conics of a harmonic system have two real common points and two real common tangents; and, if A, A' be the common points, the common tangents BC, B'C' can be so taken that AB, AC', A'B', A'C' are the tangents to the two at A, A', and the third conic of the system will touch AB, AC at B, C', and A'B', A'C' at B', C''.

1566. Prove that the equations of three conics forming a harmonic system can be obtained in areal co-ordinates in the forms

x2 = 2pyz, y2 = 2qzx, z2 = 2rxy,

where pqr + 1 = 0, the triangle of reference being either of two triangles. [By using multiples of areal co-ordinates, the equation may be written in the more symmetrical form

x2 + 2yz = 0, y2 + 2xx = 0, x2 + 2xy = 0.]

1567. In a harmonic system

x2+2yz=0, y2 + 2xx = 0, ≈2 + 2xy = 0,

(1, 2, 3)

a triangle ABC is taken whose sides touch (2) in the points a, b, c and angular points A, B, C lie on (3): prove that Aa, Bb, C'c intersect in a point lying on (3) such that if A be the point (: -A3 : λ), and B, C be similarly denoted by u, v, the point of concourse will be (: - k2 ; k) where

Also prove that be, ca, ab touch (1) in points similarly denoted by λ, μ, v. 1568. Prove that the three conics whose equations are

3x2- y2+ 4 (y+2a) 6xy=0, 3x-3y-8ay - 8a2 = 0,

form a harmonic system, which may be reduced to the standard form by either

1569. A circle and rectangular hyperbola are such that the centre of either lies on the other, and the angles at which they intersect (in real points) are 0, 6: prove that

(2 (cos 0)3 + 1) (2 (cos 0')3 + 1) = 3,

and the squares of their latera recta are as (1 + 8 cos20) : 8 sin3 0 cos 0 (the same ratio as (1 + 8 cos30') : 8 sin3 0' cos 0').

1570. A circle and parabola are such that the focus of the parabola lies on the circle and the directrix of the parabola passes through the centre of the circle, and the two intersect in two real points at angles 0, 0: prove that

(2 (cos 0)2 + 1) (2 (cos 0')3 + 1) = 3,

and that the latus rectum of the parabola is to the diameter of the circle as

8 sin3 ◊ cos: (1 + 8 cos3 0)3.

1571. A parabola and rectangular hyperbola are such that the focus of the parabola is the centre of the hyperbola and the directrix of the parabola touches the hyperbola, and they intersect in two real points at angles 20, 20': prove that

(2 (sin 0)3 + 1) (2 (sin (')3 + 1) = 3,

and that the squares of their latera recta are as

8 cos sin : (1 + 8 sin' 0)1.

THEORY OF EQUATIONS.

1572. The product of two unequal roots of the equation

1573. The roots of the equation x-px+q=0, when real, are the limits of the infinite continued fractions

1574. Prove that, when the equation 2-pa+qx-r=0 has two equal roots, the third root must satisfy either of the equations

x (x − p)2 = 4r, (x − p) (3x+p) + 49 = 0.

1575. Find the relation between p, q, r in order that the roots of the equation 3 − px2 + qx − r = 0 may be (1) the tangents, (2) the cosines, (3) the sines, of the angles of a triangle.

[The results are

(1) p=r, (2) p3 − 2q+2r = 1, (3) p1 − 4p2q +8pr + 4r2 = 0.]

1576. Prove that the roots of the equations

-5x+6x-1=0; (2) x-6x2+10x-4=0;

(1)