MATHEMATICAL QUESTIONS, WITH THEIR SOLUTIONS. FROM THE "EDUCATIONAL TIMES." WITH MANY Papers and Solutions not published in the "Educational Times." LIST OF CONTRIBUTORS. ANDERSON, D. M., Kirriemuir, Scotland. BILLS, S., Hawton, Newark-on-Trent. BLISSARD, Rev. J., B.A., The Vicarage, Hampstead Norris, Berks. BURNSIDE, W. S., B.A., Trinity College, Dublin. BROWN, JAMES, Newcastle-on-Tyne. BICKERDIKE, C., Leeds. BLAKEMORE, J. W. T., B.A., Stafford. CASEY, JOHN, B.A., Kingstown, Ireland. CAYLEY, Professor, F.R.S., University of Cambridge; Corresponding Member of the Institute of France. CLIFFORD, W. K., Trinity College, Cambridge. COCKLE, the Hon. J., M.A., Chief Justice of Queensland. COLLINS, MATTHEW, B.A., Dublin. CONWILL, J., Leighlinbridge, Ireland. CREMONA, Professor, University of Bologna. CROFTON, M. W., B.A., Royal Military Academy, Woolwich. COTTERILL, THOS., M.A., late Fellow of St. John's College, Cambridge. CONNOLLY, E., Mountnugent, Ireland. DOBSON, T., B.A., Head Master of Hexham Grammar School. DALE, JAMES, Aberdeen. EASTERBY, W., B.A., Grammar School, St. Asaph. FENWICK, STEPHEN, F.R.A.S., Royal Military Academy, Woolwich. FLOOD, P. W., Ballingarry, Ireland. FITZGERALD, E., Bagenalstown, Ireland. FULLERTON, R., Darlington. GODFRAY, HUGH, M.A., Cambridge. GODWARD, W., Law Life Office, London. GREER, H. R., B.A., Royal Military College, Sandhurst. GRIFFITHS, J., M.A., Fellow of Jesus College, Oxford. HARLEY, Rev. R., F.R.S., Brighouse, Yorkshire. HIRST, Dr. T. A., F.R.S., London. HOLDITCH, Rev. H., Caius College, Cambridge. HUDSON, W. H. H., M.A., St. John's College, Cambridge. HEYNE, J., Galway, Ireland. JENKINS, MORGAN, B.A., Christ's College, Cambridge. KIRKMAN, Rev. T. P., M.A., F.R.S., Croft Rectory, near Warrington. KNOWLES, RICHARD, London. LEVY, W. H., Shalbourne, Berks. MCCOLL, HUGH, Collège Communal, Boulogne. MCCORMICK, E., Pontrilas, Hereford. MCDOWELL, J., M.A., F.R.A.S., Pembroke College, Cambridge. MILLER, W. J., B.A., Huddersfield College. MERRIFIELD, C. W., F.R.S., Principal of the Royal School of Naval Architecture, South Kensington. MURPHY, HUGH, Killeshandra, Ireland. NELSON, R. J., M.A., London. O'CALLAGAN, J., Cahir, Ireland. O'CAVANAGH, PATRICK, Dublin. PALMER, R., M.A., London. PURKISS, H. J., B.A., Vice-Principal of the Royal School of Naval Architecture, South Kensington. RENSHAW, A., Sherwood Rise, Nottingham. RUTHERFORD, Dr., F.R.A.S., Woolwich. SADLER, G. T., F.R.A.S., London. SALMON, Rev. Dr., F.R.S., Trinity College, Dublin. SPOTTISWOODE, W., F.R.S., London. STANLEY, ARCHER. SYLVESTER, Professor, F.R.S., Royal Military Academy, Woolwich; Corresponding Member of the Institute of France. SERGEANT, LEWIS, B.A., Cheltenham. TAYLOR, C., M.A., St. John's College, Cambridge. TAYLOR, JOHN, Salop. THOMSON, F. D., M.A., Leamington College. TOWNSEND, Rev. R., M.A., Trinity College, Dublin. TUCKER, R., M.A., Newport, Isle of Wight. TODHUNTER, I., M.A., F.R.S., St. John's College, Cambridge. WALSH, R., Dingle, Ireland. WATSON, STEPHEN, Haydonbridge. WHITWORTH, W. Allen, M.A., Birkenhead. WILKINSON, T. T., F.R.A.S., Burnley. WILSON, J. M., M.A., Rugby School. WILSON, J., 45th Regiment, Colaba Camp, Bombay. WOOLHOUSE, W. S. B., F.R.A.S., &c., London. WRIGHT, Rev. R. H., M.A., Head Master of Ashford Grammar School. CONTENTS. 1058. A straight piece of wire is bent at random into two arms and then suspended by an extremity. Find the probability that the angle will, in the position of equilibrium, rise above 1160. Given the sum of two sides of a triangle, and nothing else; find the mean value of the third side 1467. n counters are marked with the numbers 1, 2, 3......n, \n (n−2) + † { 1 − (− 1)"}, or † (n−1)2 — ¿ {1+(−1)”}. 1468. Given the centre of a conic, and a conjugate triad; to con- struct for the directions of the asymptotes...... 1485. In two parallel planes (A, B) are taken m and n points respec- tively, no three of which are in the same straight line, with the exception of p of the A-points, and q of the B-points, which lie in straight lines; find (1) the number of triangles which can be formed by joining all the points in any manner, (2) the number of triangular pyramids with their bases in the 1518. A ball of weight w is projected up a smooth plane inclined at an angle a to the horizon from a given point in the plane, with a velocity B. The resistance of the air being taken to vary as the velocity of the ball, or as kv, find the position of the ball at the instant it has attained a velocity ẞ in its descent; and thence show that the point thus determined is below the point of projection.. 1521. If any odd number of terms of a geometrical progression be 1523. If [x]" denote the factorial expression x (x-1) (x-2). . (x − n + 1), 1530. Construct geometrically the expression n 2 sin(a+B) sin † (8+ y) sin † (y+a) |