4902. (C. W. Merrifield, F.R.S.)-Can a sphere be touched by more 5067. (S. Tebay, B.A.)—Let x1 + x2 + ...... + Xn = X1> Xg> ... > Xn; find the mean value of r2. = when l'+my' = 0, and r' and y' become infinite; and (2) give 5101. (A. Martin, M.A.)—An auger-hole is made through the centre 2. A circle and a rectangular hyperbola each passes through the centre of the other, and a, ẞ are the two acute angles of in- tersection of the curves at their two real common points; prove that a, ẞ will satisfy the equation (A), and that the squares of 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 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-28, prove that a, B will satisfy the equation (A), and that the squares on the latera 5146. (S. Roberts, M.A.)-Given a pencil of rays and a system of con- centric circles; prove (1) that if one set of intersections range on a straight line, the other intersections lie on a circular cubic, having a double point at the origin of the pencil and the double focus at the common centre of the circles; and (2) determine therefrom, with reference to a system of parabolas having the same focus and axis, the locus of the points the normals at which 5173. (H. T. Gerrans, B.A.)—Find the sums of the infinite series x6 x12 x18 x24 2 5 8 11 14 Page 25 x3 2.5 - 5192. (H. T. Gerrans, B.A.)-AB is a fixed diameter of a circle, OA 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 con- stant, 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 asymp- totes 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. 5224. (Rev. H. G. Day, M.A.)-On each of n pillars, whose heights, in ascending order of magnitude, are C1, C2, C3,.. are taken at random; find the chance of the point so taken on the rth pillar being the highest. 5268. (E. B. Seitz.)-Two equal circles, each of radius r, are average area common to the two circles is .(1), 5299. (L. H. Rosenthal.)-Solve the simultaneous equations, 84 .(2). 70 5304. (Professor Clifford, F.R.S.)-Prove that the negative pedal of an ellipse, in regard to the centre, has six cusps and four nodes; find their positions, and the length of the arc external to the ellipse between two real cusps; and account fully for the apparent reduction of the curve to a circle and two parabolas respectively, in special cases...... 5315. (Colonel A. R. Clarke, C.B., F.R.S.)-A straight line inter- sects a cube; show that the chance that the intercepted segment 5331. (Professor Wolstenholme, M.A.)-Prove that (1) the evolute of the first negative focal pedal of the parabola y=c(c-x) (where c = 4a= the parameter) is the curve 27 (y2-8cx- c2) 8cx (8x+9c); (2) the equation of the pedal itself is 27ay2 (3a-x) (x +6a)2; (3) the normal of the pedal exceeds the ordinate by a fixed length; (4) the arc measured from the vertex to any point is equal to the intercept of the normal on the axis of y; and (5), if a heavy uniform chain be tied tightly round a curve, such that the pressure per unit is equal to the weight of a unit of length of the chain, this curve must be the first negative focal pedal of a parabola. 5339. (Hugh McColl, B.A.) . In the quadratic equation x2+20+ y = 0, the coefficient r is taken at random between O and 3, the coefficient y between 1 and 4, and the coefficient b events will simultaneously happen-namely, that x will be the = where B is the symbol of the first Eulerian integral. 5380. (W. Gallatly, B.A.)—If a circle A touches internally another 5395. (C. Smith, M.A.)—If P, Q are points on two confocal conics, 5428. (Professor Elliott, M.A.)-Prove (1) that the highest point on the wheel of a carriage rolling on a horizontal plane moves twice as fast as each of two points in the rim whose distance from the ground is half the radius of the wheel; and (2) find the rate at which the carriage is travelling when the dirt thrown from the rim of the wheel to the greatest height attains a given level, explaining the two roots of the resulting 5436. (Dr. Booth, F.R.S.)—Express Σ (sec A) and (cosec A) in terms of the radii of circles connected with the given triangle 5437. (Christine Ladd.)—If I1, I2, I be the points of contact of the inscribed circle with the sides of a triangle ABC; 01, 02, 03 the centres of the escribed circles; ri, r, the radii of the circles inscribed in the triangles I,I,I3, О1203; and a, B, y the dis- a+b+c 5438. (The Editor.)—If A, B, C, D are four points taken at random 5441. (D. Edwardes.)-Prove that sin 2a sin (6-7)cos (-a) + sin 28 sin(-a) cos (0-8) + sin 2 sin (a-B) cos (0-y) - sin 2a sin (8—y) sin (0 − a) + sin 28 sin (y — a) sin (0—8) tan (a+B+y-0). ......... 5445. (C. H. Pillai.)-A point D is taken in the side BC produced of an equilateral triangle ABC, and CE is drawn cutting AD in E, 40 No. 5456. (Professor Minchin, M.A.)-If a body P moves in a plane orbit, so that the direction of its resultant acceleration is always a tangent to a given curve, prove that, if Q is the point of contact of this tangent, p the perpendicular from Q on the tangent at P, w the angle subtended at P by the radius of curvature at Q, the angle which PQ makes with a fixed line, and h a constant, .......... 5458. (Professor Wolstenholme, M.A.)-Find the locus of the inter- 5466. (The Editor.)-If two random points be taken one in each of ... 5467. (Christine Ladd.) — If three conics touch each other and have a common focus, prove that the common tangent of any two will cut the directrix of the third in three points which lie on one straight line. 5470. (C. W. Merrifield, F.R.S.)-Prove that broken stone, for roads, cannot weigh less per yard than half the weight of a solid yard of the same material, assuming that none of the broken faces are concave, and that it is shaken down so that there shall be no built-up hollows..... 5472. (Cecil Sharpe.)-Show how to draw a straight line terminated by the circumferences of two of three given circles, and bisected by the third circumference. 5474. (Rev. A. F. Torry, M.A.)-Find what normal divides an ellipse most unequally. construction for distance TN from ST AX: TN, directrix. About draw tangents to = 5475. (C. Taylor, M.A.)-Prove the following 5478. (C. B. S. Cavallin.)-Three straight lines are drawn at random 5479. (A. W. Panton, M.A.)-If circles be drawn on a pair of opposite sides of a square; prove that (1) the polars of any point on either diagonal with respect to the two circles meet on the other diagonal, and (2) the four tangents from the point form an harmonic pencil. No. Page 5480. (Nilkantha Sarkar, B.A.)-C is the centre of an ellipse, CB its semi-minor axis; also a circle is drawn concentric with the ellipse, and touching its two directrices, and meeting CB produced in A; determine (1) the eccentricity of the ellipse in order that CA may be bisected in B, and (2) whether the ellipse 2x2+3y2= c2 answers the condition. 5484. (A. Martin, M.A.)-Find three positive integral numbers the 5487. (Byomakesa Chakravarti.)-If SMPR be a semicircle on SR, and 5492. (M. HERMITE.)-Soit F(x) = + + 23 + 1 (n + 1) (n + 2) ' (n + 1) (n + 2) (n + 3) on demande de démontrer qu'on a F(x) F(x) 1 n2 = + n2 x2 + 24 n (n + 1)2 (n + 2) ' n (n + 1) (n + 2)2 (n + 3) (n + 4) 5496. (Professor Crofton, F.R.S.)-Prove that the mean value of the reciprocal of the distance of any two points within a circle of radius is M (1) 5497. (Professor Wolstenholme.)-A heavy uniform chain rests on a smooth arc of a curve in the form of the evolute of a catenary, a length of chain equal to the diameter of curvature at the vertex of the catenary hanging vertically below the cusp; prove (1) that the resolved vertical pressure on the curve per unit is equal to the weight of a unit-length of the chain; (2) that the resolved vertical tension is constant [the chain being, of course, fixed at its highest point]; and (3) that the former property is true for a uniform chain held tightly in contact with the curve whose intrinsic equation is s = a sin (1 + cos2 p) −3, where Φ is measured upwards from the horizontal tangent, and the directrix (or straight line from which the tension is measured) is at a depth a/2 below the vertex....... 5498. (Professor Minchin, M.A.) If E is the complete elliptic function of the second kind, with modulus k, and if k' is the complementary modulus, prove that, if n assume all values from 41 90 25 +... 76 5499. (Professor Townsend, F.R.S.)-A solid circular cylinder of uniform density and infinite length, being supposed to attract, according to the law of the inverse sixth power of the distance, a material particle projected, with the velocity from infinity under its action, from any point external to its mass, in any direction perpendicular to its axis; show that the particle will describe freely, under its action, a circular arc orthogonal to the surface of the cylinder. 55 104 92 91 |