FROM THE "EDUCATIONAL TIMES,"
Papers and Solutions not published in the "Educational Times."
Pages viii. and 27, for No. of Question 4350, read No. 4342
Page 68, last line, for (r2x2) read (r2 — x2).
Page 69, line 7 from bottom, for (R2-22) read (R2 — z2) * Page 86, line 9 from bottom, for inssde read inside.
Page 40, line 12 from bottom, for No. 133 read 134.
Page 59, line 4 from bottom, after "values" insert" of."
Page 60, line 10, for LdL read L3dL.
Page 61, line 13 from bottom, read "the respective probabilities of the promiscuous occurrence of the two cases are and ."
Page 74, line 11, for No. 136 read 137.
Page 90, line 15, for 2, 8, 6 read 2, 3, 6.
Page 99, line 12 from bottom, for No. 137 read 139. Page 106, line 10, for No. 139 read 140.
N.B. Of this series twenty-nine volumes have now been published, each volume containing, in addition to the papers and solutions that have appeared in the Educational Times, about the same quantity of new articles, and comprising contributions, on all branches of Mathematics, from most of the leading Mathematicians in this and other countries.
New Subscribers may have any of these Volumes at Subscription prices.
ALDIS, J. S., M.A.; H.M. Inspector of Schools. ALLMAN, GEO. S., LL.D.; Professor of Mathematics in the Queen's University, Galway. ARMENANTE, Professor, Pesaro. BALL, ROBT. STAWELL, LL.D., F.R.S.; Professor of Astronomy in the University of Dublin. BATTAGLINI, GIUSEPPE; Professore di Matematiche nell' Università di Roma. BELTRAMI, Professor; University of Pisa. BERG, F. J. VAN DEN; Professor of Mathematics in Delft Polytechnic School. BESANT, W. H., M.A.; Cambridge. BIRCH, Rev. J. G., M.A.; Birmingham. BLACKWOOD, ELIZABETH; Boulogne. BLISSARD, Rev. J., B.A.; The Vicarage, Hampstead Norris, Berks.
BORCHARDT, Dr. C. W.; Victoria Strasse, Berlin. BOSANQUET, R. H. M., M.A.; Fellow of St. John's College, Oxford.
BROOKS, Professor E.; Millersville, Pennsylvania. BROWN, A. CRUM, D.Sc.; Edinburgh. BROWN, COLIN, Professor in the Andersonian University, Glasgow.
BUCHHEIM, ARTHUR; New College, Oxford. BURNSIDE, W. S., M.A.; Fellow and Tutor of Trinity College, Dublin.
CAMPBELL, Capt. FRED.; Notting Hill, London. CARR, G. S.; Caius College, Cambridge. CASEY, JOHN, LL.D., F.R.S.; Prof. of Higher Mathematics in the Catholic Univ. of Ireland. CAVALLIN, C. B. S.; University of Upsala. CAVE, A. W., B.A.; Magdalen College, Oxford. CAYLEY, A., F.R.S.; Sadlerian Professor of Ma- thematics in the University of Cambridge; Member of the Institute of France, &c. CHADWICK, W.; Fellow of Christ Ch., Oxford. CHAKRAVARTI, BYOMAKESA; Presidency Col- lege, Calcutta.
CHASE, PLINY EARLE, LL.D.; Prof. of Philosophy in Haverford College.
CLARKE, Colonel A. R., C.B., F.R.S.; Director of the Ordnance Survey, Southampton. CLIFFORD, W. K., M.A., F.R.S.; late Fellow of Trinity College, Cambridge; Prof. of Applied Mathematics in University College, London. COCHEZ, Professor; Paris,
COCKLE, Hon. Sir J., Knt., M.A., F.R.S.; Chief Justice of Queensland; President of the Queensland Philosophical Society. COHEN, ARTHUR, M.A., Q.C.; London. COLSON, C. G., M.A.; University of St. Andrew's. CONSTABLE, S.; Grammar School, Drogheda. COTTERILL, J. H., M.A.; Royal School of Naval Architecture, South Kensington. COTTERILL, THOS., M.A.; London, late Fellow of St. John's College, Cambridge. CREMONA, LUIGI; Direttore della Scuola degli Ingegneri, S. Pietro in Vincoli, Rome. CROFTON, M. W., B.A., F.R.S.; Professor of Ma- thematics and Mechanics in the Royal Mili- tary Academy, Woolwich, CULVERWELL, E. P., B.A.; Scholar of Trinity College, Dublin.
DARBOUX, Professor; Paris.
DAVIS, R. F., B.A.; Wandsworth Common. DAVIS, WILLIAM BARRETT, B.A. London. DAY, Rev. H. G., M.A.; Riverside, Sevenoaks. DICK, G. R., M. A.; Fellow of Caius Coll., Camb. DOBSON, T., B.A.; Head Master of Hexham Grammar School.
DRACH, S. M.; Barnsbury Street, London. DUPAIN, J.C.; Professeur au Lycée d'Angoulême. DYER, J. M., B.A.; Cheltenham College. EASTERBY, W., B.A.; Grammar School, St. Asaph. EASTON, BELLE; Lockport, New York.
EDMUNDSON, GEORGE; Brasenose Coll., Oxford. EDWARDES, D.; Edgware. ELLIOTT, E. B., M.A.; Fellow of Qu. Coll., Oxon. ELLIS, ALEXANDER, J., F.R.S.; Kensington. ESCOTT, ALBERT, M.A.; Head Master of the Royal Hospital School, Greenwich. EVANS, Prof. A. B., M.A.; Lockport, New York. EVERETT, J. D., D.C.L.; Professor of Natural Phi-
losophy in the Queen's University, Belfast. FICKLIN, JOSEPH; Professor of Mathematics and Astronomy in the University of Missouri. FORDE, S., M.A.; Oxford. FORTEY, H., M.A.; Bellary, Madras Presidency. FRY, Colonel JOHN H.; New York. FUORTES, E.; University of Naples. GALBRAITH,J.,M.A.; FellowofTrin.Coll., Dublin, GALTON, FRANCIS, M.A., F.R.G.S.; London. GALLATLY, W., B.A.; Highgate. GARDINER, MARTIN; late Professor of Mathe- matics in St. John's College, Sydney. GENESE, R.W., M.A.; Sc. of St. John's Coll., Camb. GERRANS, H. T., B.Á.; Christ Church, Oxford. GLAISHER, J. W. L., M.A., F.R.S.; Fellow of Trinity College, Cambridge.
GLASHAN, J. C., M.A.; Strathroy, Ontario. GODFRAY, HUGH, M.A.; Newnham, Cambridge. GODWARD, WILLIAM; Chelsea. GREENFIELD, Rev. W. J., M.A.; Dulwich College. GREENWOOD, JAMES M.; Kirksville, Missouri. GRIFFITH, W.; Superintendent of Public Schools, New London, Ohio, United States. GRIFFITHS, J., M.A.; Fellow of Jesus Coll., Oxon. HALL, Professor ASAPH, M.A.; Naval Observa- tory, Washington.
HAMMOND, J., M.A.; King Edward's Sch., Bath. HARKEMA, C.; University of St. Petersburgh. HARLEY, Rev. ROBERT, F.R.S.; Vice-Master of Mill Hill Grammar School. HARRIS, H. W., B.A.; Trinity College, Dublin. HART, Dr. DAVID S.; Stonington, Connecticut, HART, H.; R.M. Academy, Woolwich. HENDRICKS, J. E., M.A.; Des Moines, Iowa. HERMITE, CH.; Membre de l'Institut, Paris. HILL, Rev. E., M.A.; St. John's College, Camb. HINTON, C. H.; Cheltenham College. HIRST, Dr. T. A., F.R.S.; Director of Studies in the Royal Naval College, Greenwich. HOPKINS, Rev. G. H.; Egham, Surrey. HOPKINSON, J., D.Sc., B.A., Manchester. HUDSON, C. T., LL.D.; Manilla Hall, Clifton. HUDSON, W. H. H., M.A.; Fellow of St. John's College, Cambridge.
INGLEBY, C. M., M.A., LL.D., London. JELLY, J. O., B.A.; Magdalen College, Oxford. JENKINS, MORGAN, M.A.; London. JENKINS, J. S.; Merton College, Oxford. JOHNSON, J. M., B.A.; Radley College, Abingdon, JOHNSON, W. W.; Annapolis, Maryland. JOHNSTON, SWIFT; Trin. Coll., Dublin. JONES, L. W., B.A.; Merton College, Oxford. KELLAND, PHILIP, M.A.; Professor of Mathe
matics in the University of Edinburgh. KING, Q. W.; Royal Hospital Sch., Greenwich. KIRKMAN, Rev. T. P., M.A., F.R.S.; Croft Rectory, near Warrington.
KITCHIN, Rev. J. L., M.A.; Heavitree, Exeter. KITTUDGE, LIZZIE A.; Boston, United States. KNISELY, Rev.U.J.; Newcomerstown, Ohio, U.S. KNOWLES, R., L.C.P.; Pentonville. LADD, CHRISTINE; Professor of Natural Sciences
and Mathematics, Union Springs, New York. LAVERTY, W. H., M.A.; Public Examiner in the University of Oxford.
LAWRENCE, E. J.; formerly Fellow of Trinity College, Cambridge.
LEIDHOLD, R., M.A. Finsbury Park. LEVETT, R., M.A.; King Edward's School, Birmingham.
LEUDESDORF, C., M.A.; Fellow of Pembroke College, Oxford.
LONG, W. S. F.; St. John's College, Cambridge. MCADAM, D. S.; Washington, Pennsylvania. MCCAY, W. S., M.A.; Fellow and Tutor of Trinity College, Dublin.
MCCOLL, HUGH; Rue Sibliquin, Boulogne. MCDOWELL, J., M.A; Pembroke Coll., Camb. MCLEOD, J., M.A.; K.M. Academy, Woolwich. MACKENZIE, J. L., B.A.; Ind. College, Taunton. MADDEN, W. M.; Trinity Parsonage, Wakefield. MALET, J. C., M.A.; Trinity College, Dublin. MANNHEIM, M.; Professeur à l'Ecole Poly- technique, Paris.
MARTIN, ARTEMAS, M.A; Editor and Publisher of the Mathematical Visitor, Erie, Pa. MARTIN, Rev. H., D.D., M.A.; Examiner in Ma- thematics in the University of Edinburgh. MATHEWS, F. C., M.A.; London. MATZ, F. P., M.A.; Beading, Pennsylvania. MERRIFIELD, C. W., F.R.S.; Brook Green. MERRIFIELD, J., LL.D., F.R.A.S.; Plymouth. MERRICK, THOS.; Kensington Square, London. MILLER, W. J. C., B.A.; 55, Netherwood Road, West Kensington Park, London, W. MINCHIN, G.M.,M.A.; Prof. in Cooper's Hill Coll. MITCHESON, T., B.A., L.C.P.; City of London Sc. MONCK, HY. STANLEY, M.A.; Prof. of Moral Phil- osophy in the University of Dublin. MONCOURT, Professor; Paris. MOON, ROBERT, M.A.; late Fellow of Queen's College, Cambridge, Cleveland Sq., London. MOREL, Professor; Paris.
MORLEY, THOS., L.C.P.; Bromley, Kent. MOULTON J. F., M.A.; Fellow of Christ's College, Cambridge.
MURPHY, HUGH; Head Master of the Incorporated Society's School, Dublin. NARENDRA LAL DEY; Presidency Coll.,Calcutta. NASH, A. M., B.A.; Professor of Nat. Phil. and
Astronomy, Presidency College, Calcutta. NELSON, R. J., M.A.; Naval School, London. O'REGAN, JOHN; New Street, Limerick. ORCHARD, H. L., B.A., L.C.P.; Hampstead. PANTON, A. W., M.A.; Fellow of Trinity College, Dublin.
PHILLIPS, F. B. W.; Balliol College, Oxford. PILLAI, C. K.; Trichy, Madras.
PIRIE, A., M.A.; University of St. Andrews. POLIGNAC, Prince CAMILLE DE; Paris. POLLEXFEN, H., B.A.; Windermere College. PRUDDEN, FRANCES E.; Lockport, New York. RAWSON, ROBERT; Havant, Hants. RENSHAW, S. A.; Nottingham. RILEY, R. E., B.A., Bournemouth. RIPPIN, CHARLES R., M.A.; Woolwich Common. ROBERTS, R. A., B.A.; Scholar of Trinity College, Dublin.
ROBERTS, SAMUEL, M.A.; Tufnell Pk., London. ROBERTS, Rev. W., M.A.; Senior Fellow Trinity College, Dublin.
ROBERTS, W. R., M.A., Ex-Scholar of Trinity College, Dublin. ROSENTHAL, L. H.; Scholar of Trin. Coll., Dublin. ROYDS, J., L.C.P.; Sheffield.
RÜCKER, A. W., B.A.; Professor of Mathematics in the Yorkshire College of Science, Leeds. RUGGERO, SIMONELLI; Università di Roma. RUTTER, EDWARD; Sunderland. SALMON, Rev. G., D.D., F.R.S.; Regius Professor of Divinity in the University of Dublin. SANDERS, J. B.; Bloomington, Indiana. SANDERSON, Rev. T. J., M.A.; Royston, Cambs. SARKAR, NILKANTHA, B.A.; Presidency Col- lege, Calcutta.
SAVAGE, THOMAS, M.A.; Fellow of Pembroke College, Cambridge.
SCHEFFER, Professor; Des Moines, Iowa, U.S. SCOTT, JOSIAH; Judge of the Ohio Supreme Court, Bucyrus, United States.
SCOTT, R. F., M.A.; Fellow of St. John's College, Cambridge.
SEITZ, E. B.; Greenville, Ohio, United States. SERRET, Professor; Paris.
SHARP, W. J. C., M.A.; Grosvenor Square. SHARPE, J. W., M.A.; The Charterhouse. SHARPE, Rev. H. T., M.A.; Cherry Marham. SHEPHERD, A. J. P.; Queen's College, Oxford. SIDES, J. J.; Rue des Vieillards, Boulogne. SIVERLY, WALTER; Oil City, Pennsylvania. SMITH, C., M.A.; Sidney Sussex Coll., Camb. SPOTTISWOODE, WILLIAM, M.A., F.R.S.;
Grosvenor Place, London. STEIN, A.; Venice. STEPHEN, ST. JOHN, B.A.; Caius Coll., Cambridge. SYMONS, E. W.; University Coll., Oxford. SYLVESTER, J. J., LL.D., F.R.S.; Professor of Mathematics in Johns Hopkins University, Member of the Institute of France, &c. TAIT, P. G., M.A.; Professor of Natural Philosophy in the University of Edinburgh. TANNER, H. W. L., M.A.; Prof. of Mathematics and Physics, R. A. College, Cirencester. TARLETON, FRANCIS A., M.A.; Fellow of Trinity College, Dublin.
TAYLOR, Rev. C., M.A.; Fellow of St. John's College, Cambridge.
TAYLOR, H. M., M.A.; Fellow and Assistant Tutor of Trinity College, Cambridge. TAYLOR, J. H., B.A.; Cambridge. TEBAY, SEPTIMUS, B.A.; Kearsley, Bolton. THOMSON, F. D., M.A.; late Fellow of St. John's
Coll., Camb., Brinkley Rectory, Newmarket. TIMOTHY, J. E.; Lockport, New York. TODHUNTER, ISAAC, F.R.S.; Cambridge. TOMLINSON, H.; Christ Church, Oxford. TORELLI, GABRIEL; University of Naples. TORRY, Rev. A. F., M.A.; St. John's College, Cambridge.
TOWNSEND, Rev. R., M.A., F.R.S.; Professor of Nat. Phil. in the University of Dublin, &c. TRAILL, ANTHONY, M.A., M.D.; Fellow and Tutor of Trinity College, Dublin. TROWBRIDGE, DAVID; Waterburgh, New York. TUCKER, R., M.A.; Mathematical Master in Uni- versity College School, London.
TURRELL, I. H.; Cumminsville, Ohio. VINCENZO, CECCHINI, University of Rome. VINCENZO, JACOBINI, University of Rome. VOSE, G. B.; Professor of Mechanics and Civil Engineering, Washington, United States. WALENN, W. H.; Mem. Phys. Society, London. WALKER, J. J., M.A.; Vice-Principal of Uni- versity Hall, Gordon Square, London. WALMSLEY, J., B.A.; Eccles, Manchester. WARD, ISABELLA M.; Capecure, Boulogne, WARREN, R., M.A.; Trinity College, Dublin. WATSON, STEPHEN; Haydonbridge. WATSON, Rev. H. W., M.A.; late Fellow of Trinity College, Cambridge.
WERTSCH, Fr.; Weimar.
WHITE, J. R., B.A.; Worcester Coll., Oxford. WHITE, Rev. J., M.A.; Woolwich. WHITWORTH, Rev. W. A., M.A.; Fellow of St.
John's Coll., Camb.; St John's Vicarage, Hammersmith.
WILKINS, W.; Scholar of Trin. Coll., Dublin. WILKINSON, Rev. M. M. U.; Norwich. WILLIAMS, S. F; Liverpool College. WILLIAMSON, B., M.A.; Fellow and Tutor of Trinity College, Dublin.
WILSON, J. M., M.A.; Rugby School. WILSON, Rev. J., M.A.; Rector of Bannockburn Academy.
WILSON, Rev. J. R., M.A.; Royston, Cambs. WILSON, Rev. R., D.D.; Chelsea. WOLSTENHOLME, Rev. J., M.A.; Professor of Mathematics in Cooper's Hill College. WOOLHOUSE, W. S. B., F.R.A.S., &c.; London. WRIGHT, Dr. S. H., M.A.; Penn Yan, New York. WRIGHT, E., B.A.; Dungannon. WRIGHT, Rev. W. J., Ph.D.; Pennsylvania. YOUNG, J. R.; London.
4253. (H. S. Monck, M.A.)-A series of Pythagorean triangles with the difference between the hypothenuse and one side equal to n, can always be obtained by beginning with the triangle 3n, 5n, 4n, and taking the upper figure as negative in each odd term of the series given in Quest. 4102. Find in what cases a distinct series with the same difference can be obtained. 23
4382. (F. C. Wace, M.A.)-At the extremities of the horizontal dia- meter of a circular wire are fixed two small rings, a third ring can slide on the wire, a string passes through the two rings and supports two weights w, w' hanging vertically; find the position of the moveable ring when it is in equilibrium..
4870. (Professor Cayley, F.R.S.)-Given three conics passing through the same four points; and on the first a point A, on the second a point B, and on the third a point C. It is required to find on the first a point A', on the second a point B', and on the third a point C', such that the intersections of the lines
A'B' and AC, A'C' and AB, lie on the first conic; B'C' and BA, B'A' and BC, lie on the second conic; C'A' and CB, C'B' and CA, lie on the third conic.....
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