For the pedal itself, we have, with the first origin, or, moving the origin from S to O, and reversing the direction of x, The curve has a crunode at the distance 9a from the vertex, the tangents there being inclined at 30° to the axis. When 0 = π, x = ·0, y = 2a, giving the point B; when = ±}π, y = 0, the node N; and if we take > π <π, giving y negative as at P, and draw PK the normal at P, = a tan 10 (2+1+tan2 10) =a (3 tan 40+ tan3 10) = arc ABP. Another obvious property of the curve is, radius of curvature varies as the square of the normal PK. In the curve in which, if a uniform chain be wrapped round it, the pressure per unit is the weight of a unit, we shall have, if be the angle which the tangent has turned through from the lowest point, the same equation as we have found for the negative pedal. The axis of the parabola must be vertical and the vertex downwards, and the string must be so wrapped on that the tension at the lowest point = kg. h weight of a length 3a of the chain. = Kg. • За = We may suppose the loop of the curve to be made into a separate curve NBA B'N, and the string wrapped round this, the tension at the highest point being four times that at the lowest. [In Reprint, Vol. XVI., p. 78, the equation of the pedal, with S as origin, is obtained in the forms (x + 4a)3 27a (x2 + y2), r = a sec3 40; and in Vol. XVII., p. 17, the curve is drawn and some of its properties investigated.] = 5212. (By Professor WOLSTENHOLME, M.A.)-A circle is drawn touching both branches of a fixed hyperbola in P, P', and meeting the asymptotes in L, L, M, M': prove that (1) LL' = MM' = major axis; (2) the tangents at L, M meet in one focus, and those at L', M' in the other, and the angle between either pair is constant, supplementary to the angle between the asymptotes; (3) the directrices bisect LM, L'M'; (4) PP' bisects LL', MM', LM, L'M'; (5) the tangents at L, L' intersect on a rectangular hyperbola passing through the foci and having one of its asymptotes coincident with MM' (because CSL + CS'L' angle between the asymptotes); (6) LM, L'M' touch parabolas having their foci at the foci of the hyperbola, and the tangents at their vertices the directrices of the hyperbola. = Solution by Professor ARMENANTE; A. MARTIN, M.A.; and others. Let the equation of the fixed and the coordinates of points (P, P') of contact will be The equation of the circle touching both branches of the hyperbola is, (4) being the equation of the asymptotes of hyperbola, we have, for the coordinates of their points of intersection, The equations of tangents to the circle at L, L' are respectively (5), (r2 — a2)3 (bx+ay + be) = a (ax—by + ac) (3·2 — a2)* (bx + ay – be) - Therefore the coordinates of the foci of the hyperbola will satisfy (5), (6); and eliminating (r2 — a2)1 from (5), (6), we have, for the locus of points of intersection of these tangents, so that (7) will be the equation of a rectangular hyperbola, one of whose asymptotes coincides with one asymptote of the fixed hyperbola (1). And since (7) is satisfied by the coordinates of the foci (±c, 0), this hyperbola passes through the foci of (1). The abscissa of the middle point of LM is -2a2c-1 ; hence one directrix bisects LM, the other bisects L'M'. = 0... (8), The equation of LM is b2 (r2 — a2) + acy (r2 — a2)* — a2 (a2 + cx) and as this equation contains the indeterminate (r2 — a2) in the 2nd degree, LM will envelop a conic, whose equation we obtain from the condition that (8) shall have equal roots; thus the envelop will be the parabola 462 (a2+ cx), or, with another origin, y2 y 462 = x. с The abscissa of the vertex, with reference to the centre of the hyperbola, is a2c-1. Therefore the tangent at the vertex of the parabola is one directrix of the hyberbola (1), and the focus of the parabola is at a distance c from the centre of the hyperbola (1); hence it coincides with one of the foci of the hyperbola (1). 5192. (By H. T. GERRANS, B.A.)—AB is a fixed diameter of a circle, OA a chord, ON an ordinate of the diameter, AP a line drawn so that = LOAN, and AP-AN; find the locus of P. Let Solution by R. E. RILEY, B.A.; E. RUTTER, and many others. OAP LOAN 0, and a = = radius of the circle; then we have AP = AN = a(1+ cos 20); therefore the locus is of the form r = a (1 + cos 4), and is, therefore, the cardioid. the OAP 5111. (By Professor WOLSTENHOLME, M.A.)-1. If a, ß be two angles such that prove that = 3 [1+2 (cos a)3] [1 + 2 (cos B)3] = ......... (A), 2. A circle and a rectangular hyperbola each passes through the centre of the other, and a, B are the two acute angles of intersection of the curves at their two real common points; prove that a, B will satisfy the equation (A), and that the squares of their latera recta are in the ratio (1 +8 cos3 a)3 : 8 sin3 a cos a .......... .(B). 3. If a circle and a parabola be such that the circle passes through the focus of the parabola, and its centre lies on the directrix, prove that their angles of intersection satisfy the equation (A), and their latera recta are in the ratio (B). 4. If a rectangular hyperbola and a parabola be such that the centre of the hyperbola is the focus of the parabola, and the directrix of the parabola touches the hyperbola; then, if their acute angles of intersection be π-2α, π-2ẞ, prove that a, 6 will satisfy the equation (A), and that the squares on the latera recta are in the ratio (B). - - [Professor WOLSTENHOLME remarks that the three curves form an harmonic system such that the polar reciprocal of any one with respect to a second is the third, and that for any two the relations = 0, '= 0 hold; so that triangles can be drawn which are inscribed to one, circumscribed to a second, and self-conjugate to the third, in any order. The equations of three such curves may always, in two ways, be reduced to the forms x5+2yz 0, y5+2zx 0, 25+ 2xy = 0.] = = Solution by the PROPOSER. 1. Let 1+2 (cos a) = k, 1+2 (cos B) = k'; then kk = 3, and or is a function of k +3k-1 only, and is unaltered by substituting 3k-1 for k, or = (1+8 cos2 6)3÷8 cos2 B sin6 B. 2. If we have a circle and rectangular hyperbola, each of which passes through the centre of the other, draw through the centre of the circle two straight lines parallel to the asymptotes of the hyperbola. These straight lines and the line at infinity will form a triangle whose angular points lie on the hyperbola, and which is self-conjugate to the circle. Similarly, if we join the centre of the hyperbola to the two points at infinity on the circle, we get a triangle inscribed in the circle and self-conjugate to the hyperbola. Hence the invariants, ' of the two curves vanish, and therefore also the equations 02 = 40′▲, '2 = 40A' are satisfied, so that we can inscribe in one triangles which either circumscribe the other or are self-conjugate to the other, and we can also circumscribe to one triangles which are either inscribed in the other or self-conjugate to it. If the equations of the two referred to the common self-conjugate triangle where w2+w+1 = 0, or m' n' m n + + - 0, or m' n' = wm w'n' is an impossible cube root of unity. The re ciprocal of either with respect to the other will then be lx2 + my2+nz2 = 0, lx2+wmy2 + w2nz2 for any pair of which the relations quences already deduced. = The harmonic locus and harmonic envelope for any two coincide with the third, so that any tangent to one is divided harmonically by the other two, and the tangents drawn from any point of one to the other two form a harmonic pencil. If we draw a parabola with its focus at the centre of the hyperbola and directrix touching the hyperbola at the centre of the circle, we see that it touches the sides of both the triangles before mentioned; and if we take the triangle formed by the asymptotes and the tangent at the centre of the circle, we see that this triangle will be inscribed to the circle, circumscribed to the hyperbola, and self-conjugate to the parabola. Hence for the parabola and either of the other curves we shall have ℗ = 0, '= 0, and the parabola will be the third curve of the system. Next, let C be a common point of the circle and rectangular hyperbola, and CA, CB tangents to the circle and hyperbola meeting the hyperbola and circle in A, B. Then, by considering the vanishing triangles circumscribed to the circle (or hyperbola), inscribed to the hyperbola (or circle), and self-conjugate to the parabola, we see that AB will touch the hyperbola and circle at A, B, and that the parabola will touch CA, CB at A, B. Similarly with the other real common point C' we shall get another triangle A'B'C' such that the circle touches A'B', A'C' at B', C', the hyperbola touches B'C', B'A' at C', A', and the parabola touches C'A', C'B' at A', B'. The equations of the three curves referred to the triangle ABC will then be of the form 0, y2+2qzx = 0, z2+2rxy = 0; but the equation of the reciprocal of the first or of the second with respect to the first, is whence we must have pqr 1. Thus the equations of the system may be written x2 +2yz = 0, y2+2zx ·0, z2+2xy = 0, and that in two ways according as we take the triangle ABC or A'B'C'. (Properly speaking, there are four ways, but only two are real.) If, however, we take areal coordinates on the triangle ABC, the three equations x2+2pyz = 0, y2+2qzx = 0, z2+2rxy will represent a circle, rectangular hyperbola, and parabola, if -a2 Where (1) again meets (2), qx3 = py3, or 8 cos1 a x3 |