4902. (C. W. Merrifield, F.R.S.)—Can a sphere be touched by more than twelve other equal spheres ? 5067. (S. Tebay, B.A.) - Let xy + x2 + ••••+ n 1, where ... > Xn; find the mean value of rm. 5090. (C. Leudesdorf, M.A.)-Evaluate (1) the equation [(ax' + hy +9) x + (hx' +by' +f)y + (gx' + fy' + c)]?, 63 of a sphere; show that the average of the volume removed is, 39 5111. (Professor Wolstenholme, M.A.) — 1. If a, B be two angles such that [1+2 (cos a)8] [1 + 2 (cos B)] = 3............ (A), (1–8 coss all_ (1 + 8 cosó Bji 2. A circle and a rectangular hyperbola each passes through the centre of the other, and a, B are the two acute angles of in- tersection 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 cos5 a) : 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 (Ā), 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 a- - 2a, – 2B, 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 6173. (H. T. Gerrans, B.A.)-Find the sums of the infinite series x 24 x3 205 X7 x9 2 5 8 11 14 3 3.5 5. 7 7.9 a chord, ON an ordinate of the diameter, AP a line drawn so that ZOAP ZOAN, and AP=AN; find the locus of P... 30 both branches of a fixed hyperbola in P, P', and meeting the x18 + + + + &c. 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, ..., Cn, points 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 16 п2. . 84 5299. (L. H. Rosenthal.)-Solve the simultaneous equations, 23— ax2 + (6 – 2y) x + ay - ( = = 0. xay -axy - (y^ — by + d) = 0.......... .(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 is less than the side of the cube is 5320. (J. J. Walker, M.A.)—If normals to the ellipse b*ir* + aạya — a«bo = 0 (a x2 + bạya — (4)3 + 54a-b°c4x*y* = 0, prove that they form an harmonic pencil. 5331. (Professor Wolstenholme, M.A.)—Prove that (1) the evolute of the first negative focal pedal of the parabola y'=c (ic – æ) 8cx (8x + 9c)”; (2) the equation of the pedal itself is 27 30° +20+ y = 0, the coefficient x is taken at random between 6 .............. events will simultaneously happen-namely, that x will be the greatest (algebraically) of the three coefficients, and that the roots of the equation will be real-is 5347. (R. Tucker, M.A.)-Prove that where B is the symbol of the first Eulerian integral. 79 respectively in a sphere, in a great circle of the sphere, in a 39 circle B at P, and a tangent to A at the point Q intersect B in 26 such that the tangents at these points are at right angles, show 23 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 E (sec A) and (cosec A) in terms of the radii of circles connected with the given triangle 5437. (Christine Ladd.)—If Iy, I2, I3 be the points of contact of the inscribed circle with the sides of a triangle ABC; O1, 02, 03 the centres of the escribed circles; ri, re the radii of the circles inscribed in the triangles 111,13, 0,0,03; and a, b, g the dis- a + b + c a + B +7 5438. (The Editor.)—If A, B, C, D are four points taken at random on the perimeter of a regular n-gon, find the respective proba- bilities that AB, CD will intersect (1) inside, (2) on, (3) outside 5441. (D. Edwardes.)-Prove that sin 2a sin (B-y)cos (6 -- a) + sin2B sin(y - a) cos(0-B) + sin 2g, sin (a- -B) cos (0-5) =tan (a + B +.7-6). 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 re 22 1 No. Page 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 55 lines; 24 35 69 47 6458. (Professor Wolstenholme, M.A.) – Find the locus of the inter section of perpendicular tangents to a cardioid, and trace the resulting curve. 5462. (Professor England, M.A.)—A variable circle passes through two given points, and through one of these pass two given find the envelope of the chord that joins the other points (1) the arcs, (2) the areas, of two semicircles that together make distance is less than the radius. have a common focus, prove that the common tangent of any on one straight line. cannot weigh less per yard than half the weight of a solid yard no built-up hollows........ 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. tangents to a conic :- Take a point T at a distance TN from will touch the conic. across a triangle ; show that the probability that cach line cuts of the triangle and a, b, c its sides. opposite sides of a square; prove that (1) the polars of any point 24 21 71 54 50 23 + + + ... + + +... 16 No. Page 6480. (Nilkantha Sarkar, B.A.)-C is the centre of an ellipse, CB its semi-minor axis ; also a circle is drawn concentric with the 41 5484. (A. Martin, M.A.)-Find three positive integral numbers the product of any two of which, diminished by the sum of the same 90 5487. (Byomakesa Chakravarti.)-If SMPR be a semicircle on SR, and the chords SP, RM cut each other in N; prove that SR2 SN. SP + RN. RM. 25 5492. (M. HERMITE.) - Soit x? F(x) = 1+ +1(n + 1)(n + 2)(n + 1) (n + 2)(n + 3) x2 76 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 55 Зп smooth arc of a curve in the form of the evolute of a catenary, a 104 5498. (Professor Minchin, M.A.) — If E is the complete elliptic function of the second kind, with modulus k, and if k' is the .3.5 ... 2n-1 92 2.4.6 2n 5499. (Professor Townsend, F.R.S.)—A solid circular cylinder of uniform density and infinite length, being supposed to attract, ........ 91 |