No. 5502. (W. S. B. Woolhouse, F.R.A.S.)—(1) Two chords are drawn in a circle; all that is known, appertaining to them, being that they intersect within the circle; determine the respective probabilities that a third random chord shall intersect neither, only one, or both of them within the circle. (2) Two chords are drawn in a circle; all that is known, appertaining to them, being that they do not intersect within the circle; determine the respective probabilities before stated. 5504. (Dr Booth, F.R.S.)—If ▲ be the area of a plane triangle, prove that 42abco, where a, b, c are the sides of the original triangle, and σ the semiperimeter of its orthocentric triangle. 5506. (The Editor.)--If A, B, C, D, E, F are six points taken at random in the perimeter of a regular n-gon, find the probability that the three intersections of (AB, CD), (CD, EF), (EF, AB) will all lie inside the perimeter. 5515. (E. P. Culverwell, M.A.)-If r be one of the four normal distances of a point P from an ellipse, and p the parallel central perpendicular on a tangent line; prove that, if {(pr)−1} vanishes, then P lies on the director circle of the ellipse; and state the corresponding theorem for an ellipsoid. 5519. (S. Roberts, M.A.)-If S and T are the fundamental invariants, 5520. (William R. Roberts, M.A.)-Show that the six inflexional Page 58 34 45 58 44 24 83 5534. (Christine Ladd.)—If four conics, S, A, B, C, have one focus 5540. (Professor Lloyd Tanner, M.A.)—If M, N are two numbers of 5544. (J. J. Walker, M.A.)-Writing HERMITE's series [Quest. 5492] x3 n F(x) = + + n (n + 1) 1 x3 + ...9 (n+3) 1. 2. 3 109 f(x) No. .......... 5548. (Elizabeth Blackwood.)-P, Q, R are three random points ......... 5553. (R. A. Roberts, B.A.)-Prove that the points of contact of parallel tangents to a Cartesian oval lie on a conic which passes through four fixed points...... Pago 7 75 86 5557. (W. H. H. Hudson, M.A.)-Find the shape of a uniform wire such that the moment of inertia of any portion of it bounded by two radii vectores about an axis through the pole perpendicular to its plane, may vary as the angle between them....... 102 5558. (C. Leudesdorf, M.A.)—Given a number of equal-sized spheres in the closest possible contact; find how many each would touch. 5563. (Professor Sylvester, F.R.S.)-If at two points in a cubic curve, lying in a straight line with a point of inflexion, tangents be drawn to meet the curve again, prove that their intersections with it will also be in a straight line with the same point of inflexion. .............. 85 .52, 93 5564. (Professor Benjamin Peirce, F.R.S.)-Find the probabilities ........ 72 41 ......... 5571. (Christine Ladd.)-Find the locus of a point with regard to which the reciprocal of a fixed triangle has a constant area, provided the radius of the auxiliary circle remains constant. 5572. (Elizabeth Blackwood.)-The centre of a given circle is O, and P and Q are two random points within the circle; find the chance that the triangle POQ is less than a given area. 5573. (Professor Monck, M.A.) The three edges and the diagonal of a rectangular parallelepiped are integer numbers; show how to obtain a series of parallelepipeds possessing the same quality. ... 65 106 74 5575. (The Editor.)-If of four triangles A,B,C1, A,B,С2, A,B,C, A4B4C4, the first is in perspective with the second, the second with third, the third with fourth, and the fourth with first, in such a way that the vertices of the same letters are corresponding; and if the four centres of perspective lie in a straight line; prove that the four perspective lines meet in a point...... 53 No. 5580. (S. Roberts, M.A.)—If the sides of a variable triangle pass through three fixed points in a straight line, while one vertex moves on another straight line, and a second vertex describes a given curve, prove that the locus of the third vertex is a homographic transformation of the given curve. ............... 5586. (R. E. Riley, B.A.)--If a series of circles be described concentric with an ellipse of eccentricity e, prove that the chords of contact of tangents drawn to the ellipse from points in these circles envelop a series of concentric similar ellipses of eccentricity (2-e2). e Page 96 46 5588. (T. Mitcheson, B.A., L.C.P.)-If on each side of a triangle ABC triangles similar thereto are drawn, so that the angles adjacent to A are each equal to C, those adjacent to B each equal to A, and those adjacent to C each equal to B: prove that three circles drawn round the outer triangles, and the three lines that join their vertical angles with the opposite angles of ABC, all intersect at a point O, such that the angles OBC, OCA, OAB are equal to each other; and that cot OBC = cot A + cot B+ cot C. 98 5590. (S. A. Renshaw.)-If a quadrilateral be inscribed in a conic, show that a point may be found on each of its sides such that, the four points being joined, a quadrilateral inscribed in the former will be formed whose opposite sides produced will meet on the directrix. 103 5594. (Professor Townsend, F.R.S.)-Prove the following pairs of reciprocal properties of a system of two conics : (a) When two conics are such that two of their four common points subtend harmonically the angle determined by the tangents at either of the remaining two, they subtend harmonically that determined by those at the other also. (b) When two conics are such that two of their four common tangents divide harmonically the segment determined by the points of contact of either of the remaining two, they divide harmonically that determined by those of the other also. (c) The associated conic, envelope of the system of lines divided harmonically by the two original conics, breaks up, in the former case, into the point-pair determined by the eight tangents to them at their four common points. (d) The associated conic, locus of the system of points 5599. (The Editor.)-Calling the straight line that passes through = .(1), (o2 — a2) l ̧2 + (σ2 — b2) 11⁄22 + (σ2 — c2) 132 = (28)2 ................................ (2), 88 (o2 — a2) a2. AP2+(o2 — b2) b2 . BP2 + (o2— c2) c2 . CP2 = a2b2c2 .....(3). 80 No. Page 5600. (Christine Ladd.)-Required the envelop of the SIMSON line.... 80 5602. (Professor Monck, M.A.)-Two chords of given length are drawn at random within a given circle; find the chance that they will intersect within the circle. 5605. (J. C. Malet, M.A.) If a quadric V intersects another quadric U in the planes L and M, and passes through the pole of L with respect to U; prove that it will also pass through the pole of M with respect to U. 85 103 5606. (R. E. Riley, B.A.)-A particle slides down a rough parabola whose axis is vertical, starting from an extremity of the latusrectum. If it stops at the vertex, prove that μ = T-1 loge 4... 108 5607. (J. Royds, A.C.P.)-Find x from the equation (a−x)3 + (a−x) = b. 5609. (A. W. Panton, M.A.)-The four common tangents to two circles being supposed such that the two opposite pairs, external and internal, are at right angles to each other; show that their eight points of contact with the circles lie on two straight lines, every point on each of which subtends the circles in an harmonic system of tangents. 5610. (J. Hammond, M.A.)-Prove that 5611. (Professor Wolstenholme, M.A.)—Having given that 87 101 79 e (y2x2 + 1) + (y2 + z2) _ € (z2x2 + 1) + (x2 + x2) _ e (x2y2 + 1) + (x2 + y2) = k2 yz = prove that k = e2-1, and that yz + zx + xy (yz) -1+(zx)−1 + (xy)−1. = xy 5613. (R. A. Roberts, M.A.)-Prove that the locus of the centroid of a triangle inscribed in a conic and circumscribed to a parabola is a straight line......... 5621. (D. Edwardes.)-If a circle be drawn through the centre of the inscribed circle and the centres of any two escribed circles of a triangle, prove that its radius is double that of the circumscribed circle of the triangle. 5623. (C. K. Pillai.)—ABCD is a parallelogram; a point E is taken in the diagonal BD, and a point F in CE; also FG is drawn parallel to DC meeting the diagonal in G, and GH is drawn from G parallel to BF, and meeting AB in H; prove that AH: AB: DG: DE. = 5625. (Professor Cayley, F.R.S.)-The equation {q2 (x + y + z)2 — yz-zx-xy} = 4 (2q+1) xyz (x + y + z) represents a trinodal quartic curve having the lines x=0, y=0, z=0,x+y+z = 0 for its four bitangents; it is required to transform to the coordinates X, Y, Z, where X=0, Y=0, Z=0 represent the sides of the triangle formed by the three nodes. 5654. (T. Mitcheson, B.A., L.C.P.)—If within a triangle a point O be taken such that ACO BAO CBO = 0, prove that cot e = = - cot A+ cot B+ cot C....... = k, 100 99 73 109 96 98 MATHEMATICS FROM THE EDUCATIONAL TIMES. WITH ADDITIONAL PAPERS AND SOLUTIONS. NOTES ON RANDOM CHORDS. By the EDITOR. Since the publication of Miss BLACKWOOD's Solution of Question 5461, with Mr. WOOLHOUSE's notes thereon (Reprint, Vol. XXVIII., pp. 108-110), we have received several communications on the subject of random chords, from which we now select for publication the following remarks: 1. Colonel CLARKE's views on the subject are expressed as follows:— A question very similar to 5461 was proposed by Mr. WOOLHOUSE in the Lady's and Gentleman's Diary for 1856, in the following terms:-"In a dark room two persons each of them draw a chord at random across a circular slate; what is the chance that they will intersect ?" The solutions adopted at the time gave as the answer; therein agreeing with Miss BLACKWOOD's. But there is another solution which gives as the chance; and these are the solutions of these problems : (A) "Two random points P1, P2 on the circumference of a circle are joined by a straight line, and two other random points Q1, Q2 are joined by a straight line; what is the chance that these lines intersect within the circle?" The answer here is, without question, . Or, expressed otherwise, thus:-(A) "Through each of two random points P, Q on the circumference of a circle, a chord is drawn in any direction at random, what is the chance that these chords intersect within the circle ?" (B) "Two lines are drawn at random across a circle, what is the chance that they intersect within the circle ?" The answer here is, without question, In (A) the distribution of lines or chords is that of lines drawn through random points on the circumference of a circle. But such lines are not random lines; since for a given direction they lie closer together towards the circumference, and are further apart at the centre; they are not random lines, or they could not have this particular law of density. In Question 5461, the expression is "all possible chords being supposed equally probable." Here we must suppose an even distribution of chords. In problem (A) the chords are distributed evenly as to direction, but not as to distance from the centre; and we might evidently have even distribution as to direction combined with any arbitrary law of distribution as |