MATHEMATICAL QUESTIONS, WITH THEIR SOLUTIONS, FROM THE "EDUCATIONAL TIMES," WITH MANY Papers and Solutions not published in the "Educational Times." EDITED BY W. J. C. MILLER, B.A., VICE PRINCIPAL OF HUDDERSFIELD COLLEGE. UNIVERSITY OF MINNESOTA .... FROM JULY TO DECEMBER, 1873. LONDON: C. F. HODGSON & SONS, GOUGH SQUARE, FLEET STREET. 1874. bov. 7280 LIST OF CONTRIBUTORS. ALDIS, J. S., M.A., H.M. Inspector of Schools. BATES, GILL, Prescott, Iowa, United States. BIRCH, Rev.J.G., M.A., Royal Military Academy, BLACKWOOD, ELIZABETH, London. BLISSARD, Rev. J., B.A., The Vicarage, Hampstead Norris, Berks. BOOTH, Rev. Dr., F.R.S., The Vicarage, Stone, near Aylesbury, Bucks. BOURKE, Hon. ALGERNON, Berkeley Square, London. BOWDITCH, W. L., Wakefield. BRANQUART, PAUL, Church House, Ealing. CARNT, E. C., Royal Hospital School, Greenwich. CHADWICK, W., Fellow of Corpus Christi College, Oxford. CLARKE, Colonel, C.B., F.R.S., Ordnance Survey Office, Southampton. CLARKE, Miss M. A., Birmingham. Justice of Queensland; President of the COHEN, ARTHUR, M.A., London. CROFTON, M. W., B.A., F.R.S., Professor of a thematics and Mechanics in the Royal Mili- DUPAIN, J. C., Professeur au Lycée d'Angoulême. EVERETT, J. D., D.C.L., Professor of Natural Phi- FITZGERALD, E., Bagenalstown, Ireland. FUORTES, E., Etudiant à l'Université de Naples. GARDINER, MARTIN, late Professor of Mathematics in St. John's College, Sydney. GENESE, R. W., B.A., Scholar of St. John's Coll., Cambridge, Vice-Principal of South Wales Training College. GLAISHER, J. W. L., B.A., Fellow of Trinity College, Cambridge. GODFRAY, HUGH, M.A., Cambridge. GREENWOOD, JAMES M., Kirksville, Missouri, GRIFFITH, W., Superintendent of Public Schools, HANNA, W., Literary Institute, Belfast. HERMITE, Ch., Membre de l'Institut, Paris. HOPKINS, Rev. . H., M.A., Shrewsbury Road, HOSKINS, HENRY, Alma Place, Kensington. INGLEBY, C. M., M.A., LL.D., Royal Society of KIRKMAN, Rev. T. P., M.A., F.R.S., Croft Rectory, near Warrington. KITCHENER, F. E., M.A., Rugby School. KITCHIN, Rev. J. L., M.A., Head Master of Bideford Grammar School. KITTUDGE, LIZZIE A., Boston, United States. KNISELY, Rev. U.J., Newcomerstown, Ohio, U.S. LADD, CHRISTINE, Poynonock, Connecticut. та LAVERTY, W. H., M.A., late Fellow of Queen's College, and Public Examiner in the Univ. of Oxford; Headly Rectory, Hants. LEUDESDORF, C., B.A., Pembroke Coll., Oxford. MCCAY, W. S., B.A., Trinity College, Dublin. MCCOLL, HUGH, Rue du Temple, Boulogne-surMer, France. MCCORMICK, E., Ledbury, Hereford. MCDOWELL, J., M.A., F.R.A.S., Pembroke College, Cambridge. MCLEAN, ABIJAH, New Lisbon, Ohio, U.S. MCLEOD, J., M.A., Instructor in Mathematics in the Royal Military Academy, Woolwich. MACKENZIE, J. L., Clifton Bank School, St. Andrews. MADDEN, W. M., Trinity Parsonage, Wakefield. MALET, J. C., M.A., Trinity College, Dublin. MANNHEIM, M., Professeur à l'Ecole Polytechnique, Paris. iv MARTIN, ARTEMAS, Mathematical Editor of "The Visitor," Erie, Pennsylvania, U. S. MARTIN, Rev. H., D.D., M.A., Examiner in Mathematics in the University of Edinburgh. MASON, J., East Castle Colliery, near Newcastleon-Tyne. MATHEWS, F. C., M.A., London. MERRIFIELD, C. W., F.R.S., Pembroke Gardens, MERRIFIELD, J., LL.D., F.R.A.S., Science School, Plymouth. MERRICK, THOS., Kensington Square, London. MILLER, W. J. C., B.A., 5, Brunswick Street, New North Road, Huddersfield. MINCHIN, G. M., M.A., Trinity College, Dublin. MITCHESON,T.,B.A.,The Cedars, Rickmansworth. MOON, ROBERT, M.A., late Fellow of Queen's College, Cambridge, Cleveland Sq., London. MONCK, HENRY STANLEY, Trinity Coll., Dublin. MORLEY, THOS., L.C.P., Bromley, Kent. MOULTON, J. F., M.A., Fellow of Christ's College, Cambridge. MULCASTER, J., Woolwich Common. O'CAVANAGH, PATRICK, Dublin. OGILVIE, G. A., Leiston, Saxnfungtham.. PANTON, A. W., B.A., Trinity College, Dublin. PRUDDEN, FRANCES E. Lockport, New Yorky United States. RENSHAW, S. A., Observatory, Cambridge.. RIPPIN, CHARLES R., M.A., Woolwich Common. SANDERS, J. B., Bloomington, Monroe County, SAVAGE, THOMAS, M.A., Fellow of Pembroke SCOTT, JOSIAH, Judge of the Ohio Supreme SHARPE, Rev. H. T., M.A., Vicar of Cherry Marham, Norfolk. SIDES, J. J., Rue des Vieillards, Boulogne-surMer, France. SIVERLY, WALTER, Oil City, Pennsylvania, U.S. SPRAGUE, THOMAS BOND, M.A., London. SYMES, R. W., B.A., London. TAYLOR, Rev. C., M.A., Fellow of St. John's Coll., TAYLOR, H. M., M.A., Fellow and Assistant Tutor of Trinity College, Cambridge. TAYLOR, J. H., B.A., Cambridge. TEBAY, SEPTIMUS, B.A., Head Master of Rivington Grammar School. THOMSON, F. D., M.A., late Fellow of St. John's Coll., Camb., Brinkley Rectory, Newmarket. TIMOTHY, J. E., Lockport, New York, U.S. TODHUNTER, ISAAC, F.R.S., St. John's College, Cambridge. TOMLINSON, H., Christ Church College, Oxford. Trinity College, and Professor of Natural Philosophy in the University of Dublin. TRAILL, ANTHONY, M.A., M.D., Fellow of Trinity College, Dublin. TROWBRIDGE, DAVID, Waterburgh, New York, TUCKER, R., M.A., late Scholar of St. John's Col. WALMSLEY, J., B.A., Eccles, Manchester. WARREN, R., M.A., Trinity College, Dublin. WATSON, STEPHEN, Haydonbridge, Northumberland. WE, Rev. J., M.A., Woolwich. WHITWORTH, Rev. W. A., M.A., Fellow of St. WILKINSON, Rev. M. M. U., Reepham Rectory, ...Norwich. WILKINSON, T. T., F.R.A.S., Burnley. WILLIAMSON, B., M.A., Fellow of Trinity College, Dublin. WILSON, J. M., M.A., Nat. Phil. Master in Rugby School. WILSON, Rev. R., D.D., Chelsea. WOLSTENHOLME, Rev. J., M.A., late Fellow of Christ's College, Cambridge; Professor of Mathematics in the Civil Engineering College, Cooper's Hill, Staines. WOOLHOUSE, W. S. B., F.R.A.S., &c., Alwyne Lodge, Canonbury, London. WORDINGHAM, ELIZABETH, Fakenham. WRIGHT, W. E., Royal School, Raphoe. CONTENTS. Mathematical Papers, &c. (2) that no integral values of x can be found to satisfy the 2981. (Rev. G. H. Hopkins, M.A.)-If the sum of the squares of two 63 3069. (The Editor.)-Find the sides of a triangle in rational num- 56 3115. (Dr. Hart.)-To find five biquadrate numbers whose sum is 55 |