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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.,

REGISTRAR OF THE GENERAL MEDICAL COUNCIL.

BLIOTHER

VOL. XXIX. 1. OCT '578 )
FROM JANUARY TO JUNE, 1874.

PODLELANEY

LONDON:
C. F. HODGSON & SON, GOUGH SQUARE,

· FLEET STREET.

1878.

18753.2.2.

CORRIGENDA.

VOL. XXII.
Pages viii. and 27, for No. of Question 4350, read No. 4342

VOL. XXVIII,
Page 68, last line, for (v2.x?) read (72 — x2).
Page 69, line 7 from bottom, for (R2–22) read (R2— 22)
Page 86, line 9 from bottom, for inssde read inside.

VOL. XXIX.
Page 41, line 12 from bottom, for No. 133 read 134.
Page 59, liue 4 from bottom, after " values” insert “ of."
Page 60, line 10, for LdL read LdL.

Page 61, line 13 from bottom, read the respective probabilities of the promiscuous occurrence of the two cases are and 3."

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.

LIST OF CONTRIBUTORS.

ALDIS, J. S., M.A.; H.M. Inspector of Schools. EDMUNDSON, GEORGE; Brasenose Coll., Oxford. ALLMÁN, GEO. S., LL.D.; Professor of Mathe- EDWARDES, D.; Edgware.

matics in the Queen's University, Galway. ELLIOTT, E. B., M.A.; Fellow of Qu. Coll., Oxon, ARMENANTE, Professor, Pesaro.

ELLIS, ALEXANDER, J., F.R.S.; Kensington. BALL, ROBT. STAWELL, LL.D., F.R.S.; Professor ESCOTT, ALBERT, M.A.; Head Master of the

of Astronomy in the University of Dublin. Royal Hospital School, Greenwich. BATTAGLINI, GIUSEPPE; Professore di Mate- EVANS, Prof. A. B., M.A.; Lockport, New York. matiche nell' Università di Roma.

EVERETT,J.D., D.C.L.; Professor of Natural PhiBELTRAMI, Professor, University of Pisa.

losophy in the Queen's University, Belfast. BERG, F.J. VAN DEN; Professor of Mathematics FICKLIN, JOSEPI; Professor of Mathematics and in Delft Polytechnic School.

Astronomy in the University of Missouri. BESANT, W. H., M.A.; Cambridge.

FORDE, S., M.A.; Oxford. BIRCI, Rev. J. G., M.A.; Birmingham.

FORTEY, H., M.Á. ; Bellary, Madras Presidency. BLACKWOOD, ELIZABETH; Boulogne.

Fry, Colonel JOHN H.; New York. BLISSARD, Rev. J., B.A.; The Vicarage, Hamp- FUORTES, E.; University of Naples. stead Norris, Berks.

GALBRAIT 11,J.,M.A.; FellowofTrin.Coll., Dublin. BORCHARDT, Dr. C. W.; Victoria Strasse, Berlin. GALTON, FRANCIS, M.A., F.R.G.S.; London, BOSANQUET, R. H. M., M.A.; Fellow of St. GALLATLY, W., B.A.; Highgate. John's College, Oxford.

GARDINER, MARTIN; late Professor of Mathe. BROOKS, Professor E.; Millersville, Pennsylvania. matics in St. John's College, Sydney. BROWN, A. CRUM, D.Sc.; Edinburgh.

GENESE, R. W.,M.A.; Sc. of St. John's Coll.,Camb. BROWN, COLIN, Professor in the Andersonian GERRANS, H. T., B.$. ; Christ Church,

Oxford. University, Glasgow.

GLAISHER, J. W. L., M.A., F.R.S.; Fellow of BUCHHEIM, ARTHUR; New College, Oxford. Trinity College, Cambridge. BURNSIDE, W. S., M.A.; Fellow and Tutor of GLASHAN, J.C., M.A.; Strathroy, Ontario. Trinity College, Dublin.

GODFRAY, HUGH, M.A.; Newnham, Cambridge. CAMPBELL, Capt. FREV.; Notting Hill, London. GODWARD, WILLIAM ; Chelsea. CARR, G. S.; Caius College, Cambridge.

GREENFIELD, Rev. W.J., M.A.; Dulwich College. CASEY, JOHN, LL.D., F.R.S.; Prof. of Higher GREENWOOD, JAMES M.; Kirksville, Missouri. Mathematics in the Catholic Univ. of Ireland. GRIFFITH, W.; Superintendent

of Public Schools, CAVALLIN, C. B. S.; University of Upsala.

New London, Ohio, United States. CAVE, A. W., B.A.; Magdalen College, Oxford. GRIFFITHS, J., M.A.; Fellow of Jesus Coll., Oxon. CAYLEY, A., F.R.S.; Sadlerian Professor of Ma- HALL, Professor ASAPH, M.A.; Naval Observa

thematics in the University of Cambridge ; tory, Washington.

Member of the Institute of France, &c. HAMMOND, J., M.A.; King Edward's Sch., Bath, CHADWICK, W.; Fellow of Christ Ch., Oxford. HARKEMA, C.; University of St. Petersburgh. CHAKRAVARTI, BYOMAKESA ; Presidency Col. HARLEY, Rev. ROBERT, F.R.S.; Vice-Master of lege, Calcutta.

Mill Hill Grammar School, CHASE, PLINY EARLE, LL.D.; Prof. of Philosophy HARRIS, H. W., B.A.; Trinity College, Dublin. in Haverford College.

HART, Dr. DAVID S.; Stonington, Connecticut, CLARKE, Colonel A.R., C.B., F.R.S.; Director of HART, H.; R.M. Academy, Woolwich.

the Ordnance Survey, Southampton. HENDRICKS, J. E., M.A.; Des Moines, Iowa. .CLIFFORD, W. K., M.A., F.R.S.; late Fellow of HERMITE, CA.; Membre de l'Institut, Paris.

Trinity College, Cambridge; Prof. of Applied HILL, Rev. E., M.A.; St. John's College, Camb.

Mathematics in University College, London. HINTON, C. H.; Cheltenham College. COCHEZ, Professor; Paris.

HIRST, Dr. T. A., F.R.S.; Director of Studies in COCKLE, Hon. Sir J., Knt., M.A., F.R.S.; Chief the Royal Naval College, Greenwich.

Justice of Queensland; President of the HOPKINS, Rev. G. H.; Egham, Surrey.
Queensland Philosophical Society.

HOPKINSON, J., D.Sc., B.A., Manchester.
COHEN, ARTHUR, M.A., Q.C.; London.

Hudson, C. T., LL.D.; Manilla Hall, Clifton. COLSON, C. G., M.A.; University of St. Andrew's. HUDSON, W. Ú. H., M.A.; Fellow of St. John's CONSTABLE, S.; Grammar School, Drogheda. College, Cambridge. COTTERILL, J. H., M.A.; Royal School of Naval INGLEBY, C. M., M.A., LL.D., London. Architecture, South Kensington.

JELLY, J.O., B.A.; Magdalen College, Oxford, COTTERILL, THos., M.A.; London, late Fellow of JENKINS, MORGAN, M.A.; London. St. John's College, Cambridge.

JENKINS, J. S.; Merton College, Oxford. CREMONA, LUIGI; Direttore della Scuola degli JOHNSON, J. M., B.A.; Radley College, Abingdon,

Ingegneri, S. Pietro in Vincoli, Rome. Johnson, W. W.; Annapolis, Maryland. CROFTON, M. W., B.A., F.R.S. ; Professor of Ma. JOHNSTON, SWIFT ; Trin. Coll., Dublin.

thematics and Mechanics in the Royal Mili- JONES, L. W., B.A.; Merton College, Oxford. tary Academy, Woolwich.

KELLAND, PHILIP, M.A.; Professor of Mathe. CULVERWELL, E. P., B.A.; Scholar of Trinity matics in the University of Edinburgh. College, Dublin.

KING, Q. W.; Royal Hospital Sch., Greenwich. DARBOUX, Professor; Paris.

KIRKMĂN, Rev. T. P., M.A., F.R.S. ; Croft RecDAVIS, R. F., B.A.; Wandsworth Common.

tory, near Warrington, Davis, WILLIAM BARRETT, B.A. London. KITCHIN, Rev. J. L., M.A.; Heavitree, Exeter. Day, Rev. H. G., M.A.; Riverside, Sevenoaks. KITTUDGE, LIZZIE A. ; Boston, United States. Dick, G. R., M. A.; Fellow of Caius Coll., Camb. KNISELY,Rev.U.J.; Newcomerstown, Ohio, U.S. DOBSON, T., B.A.; Head Master of Hexham KNOWLES, R., L.C.P. ; Pentonville. Grammar School.

LADD, CHRISTINE; Professor of Natural Sciences DRACH, S. M. ; Barnsbury Street, London. and Mathematics, Union Springs, New York, DUPAIN, J.C. Professeur au Lycée d'Angoulênie. LAVERTY, W. H., M.A.; Public Examiner in the DYER, J. M., B.A.; Cheltenham College.

University of Oxford. EASTERBY, W., B.A.; GrammarSchool, št. Asaph. LAWRENCE, E. J.; formerly Fellow of Trinity EASTON, BELLE; Lockport, New York.

College, Cambridge.

iv

LEIDHOLD, R., M.A. Finsbury Park.

Scott, R. F., M.A.; Fellow of St. John's College, LEVETT, R., M.A.; King Edward's School, Cambridge. Birmingham.

Seitz, E. B. ; Greenville, Ohio, United States. LEUDESDORF, C., M.A.; Fellow of Pembroke SERRET, Professor ; Paris. College, Oxford.

SHARP, W. J. C., M.A.; Grosvenor Square. LONG, W. S. F. ; St. John's College, Cambridge. SHARPE, J. W., M.A.; The Charterhouse. McAdam, D. S.; Washington, Pennsylvania. SHARPE, Rev. H. T., M.A. ; Cherry Marham. McCAY, W. S., M.A.; Fellow and Tutor of SHEPHERD, A. J. P.; Queen's College, Oxford. Trinity College, Dublin.

SIDES, J. J.; Rue des Vieillards, Boulogne. McCOLL, HUGH; Rue Sibliquin, Boulogne. SIVERLY, WALTER; Oil City, Pennsylvania. MCDOWELL, J., M.A ; Pembroke Coll., Camb. SMITI, C., M.A. ; Sidney Sussex Coll., Camb. MCLEOD, J., M.A.; R.M. Academy, Woolwich. SPOTTISWOODE, WILLIAM, M.A., F.R.S.; MACKENZIE, J. L., B.A.; Ind. Coilege, Taunton. Grosvenor Place, London. MADDEN, W. M.; Trinity Parsonage, Wakefield. STEIN, A.; Venice. MALET, J. C., M.A.; Trinity College, Dublin. STEPHEN, ST.JOHN,B.A.; Caius Coll., Cambridge. MANNHEIM, M.; Professour à l'École Poly- Symons, E. W., University Coll., Oxford. technique, Paris.

SYLVESTER, J.J., LL.D., F.R.S.; Professor of MARTIN, ARTEMAS, M.A ; Editor and Publisher Mathematics in Johns Hopkins University, of the Mathematical Visitor, Erie, Pa.

Member of the Institute of France, &c. MARTIN, Rev. H., D.D., M.A.; Examiner in Ma- TAIT, P. G., M.A.; Professor of Natural Philo

thematics in the University of Edinburgh. sophy in the University of Edinburgh. MATHEWS, F. C., M.A.; London.

TANNER, H. W. L., M.A. ; Prof. of Mathematics Matz, F. P., M.A.; Reading, Pennsylvania.

and Physics, R. A. College, Cirencester. MERRIFIELD, C. W., F.R.S.; Brook Green. TARLETON, FRANCIS A., M.A. ; Fellow of Trinity MERRIFIELD, J., LL.D., F.R.A.S.; Plymouth. College, Dublin. MERRICK, THOS.; Kensington Square, London. TAYLOR, Rev. C., M.A.; Fellow of St. John's Col. MILLER, W. J. C., B.A.; 55, Netherwood ad,

Cambridge. West Kensington Park, London, W.

TAYLOR, H.M., M.A.; Fellow and Assistant Tutor MINCHIN, G.M.,M.A.; Prof. in Cooper's Hill Coll. of Trinity College, Cambridge. MITCHESON, T., B.A., L.C.P.; City of London Sc. TAYLOR, J. İ., B.A.; Cambridge. MONCK, HY. STANLEY, M.A. ; Prof. of Moral Phil. TEBAY, SEPTIMUS, B.A.; Kearsley, Bolton. osophy in the University of Dublin.

THOMSON, F. D., M.A. ; late Fellow of St. John's MONCOURT, Professor ; Paris.

Coll., Camb., Brinkley Rectory, Newmarket. MOON, ROBERT, M.A.; late Fellow of Queen's TIMOTHY, J. E. ; Lockport, New York.

College, Cambridge, Cleveland Sq., London. TODHUNTER, ISAAC, F.R.S. ; Cambridge. MOREL, Professor; Paris.

TOMLINSON, H.; Christ Church, Oxford. MORLEY, Thos., L.C.P.; Bromley, Kent. TORELLI, GABRIEL; University of Naples. MOULTON J. F., M.A.; Fellow of Christ's College, TORRY, Rev. A. F., M.A.; St. John's College, Cambridge.

Cambridge. MURPHY, HUGH; Head Master of the Incor- TOWNSEND), Rev. R., M.A., F.R.S.; Professor of porated Society's School, Dublin.

Nat. Phil. in the University of Dublin, &c. NARENDRA LAL DEY; Presidency Coll.,Calcutta. TRAILL, ANTHONY, M.A., M.D.; Fellow and NASH, A. M., B.A. ; Professor of Nat. Phil. and Tutor of Trinity College, Dublin.

Astronomy, Presidency College, Calcutta. TROWBRIDGE, DAVID ; Waterburgh, New York. NELSON, R. J., M.A.; Naval School, London. TUCKER, R., M.A.; Mathematical Master in UniO’REGAN, JOHN; New Street, Limerick.

versity College School, London. ORCHARD, H. L., B.A., L.C.P. ; Hampstead. TURRELL, I. H.; Cumminsville, Ohio. Panton, Á. W., M.A. ; Fellow of Trinity College, VINCENZO, CECCHINI, University of Rome. Dublin.

VINCENZO, JACOBINI, University of Rome. PHILLIPS, F. B. W.; Balliol College, Oxford. VOSE, G. B.; Professor of Mechanics and Civil PILLAI, C. K.; Trichy, Madras.

Engineering, Washington, United States. PIRIE, A., M.A.; University of St. Andrews. WALENN, W. H.; Mem. Phys. Society, London. POLIGNAC, Prince CAMILLE DE; Paris.

WALKER, J. J., M.A.; Vice-Principal of UniPOLLEXFEN, H., B.A.; Windermere College. versity Hall, Gordon Square, London. PRUDDEN, FRANCES E.; Luckport, New York. WALMSLEY, J., B.A. ; Eccles, Manchester. Rawson, ROBERT; Havant, Hants.

WARD, ISABELLA M.; Capecure, Boulogne, RENSHAW, S. A.; Nottingham.

WARREN, R., M.A.; Trinity College, Dublin. RILEY, R. E., B.A., Bournemouth.

WATSON, STEPHEN ; Haydonbridge. RIPPIN, CHARLES R., M.A.; Woolwich Common. WATSON, Rev. H. W., M.A.; late Fellow of Trinity ROBERTS, R. A., B.A.; Scholar of Trinity College, College, Cambridge. Dublin.

WERTSCH, Fr.; Weimar. ROBERTS, SAMUEL, M.A.; Tufnell Pk., London. WHITE, J. R., B.A.; Worcester Coll., Oxford. ROBERTS, Rev. W., M.A.; Senior Fellow Trinity WHITE, Rev. J., M.A.; Woolwich. College, Dublin.

WHITWORTH, Rev. W. A., M.A.; Fellow of St. ROBERTS, W. R., M.A., Ex-Scholar of Trinity John's Coll., Camb.; St John's Vicarage, College, Dublin.

Hammersmith. ROSENTHAL, L. H.; Scholar of Trin. Coll., Dublin. WILKINS, W.; Scholar of Trin. Coll., Dublin. ROYDS, J., L.C.P.; Sheffield.

WILKINSON, Rev. M. M. U.; Norwich. RÜCKER, A. W., B.A.; Professor of Mathematics WILLIAMS, S. F ; Liverpool College.

in the Yorkshire College of Science, Leeds. WILLIAMSON, B., M.A.; Fellow and Tutor of RUGGERO, SIMONELLI; Università di Roma. Trinity College, Dublin. RUTTER, EDWARD; Sunderland.

WILSON, J.M., M.A.; Rugby School. SALMON, Rey. G., D.D., F.R.S.; Regius Professor Wilson, Rev. J., M.A.; Rector of Bannockburn of Divinity in the University of Dublin.

Academy. SANDERS, J. B.; Bloomington, Indiana.

WILSON, Rev. J. R., M.A.; Royston, Cambs. SANDERSON, Rev.T. J., M.A.; Royston, Cambs. Wilson, Rev. R., D.D.; Chelsea. SARKAR, NILKANTHA, B.A.; Presidency Col- WOLSTENHOLME, Rev. J., M.A.; Professor of lege, Calcutta.

Mathematics in Cooper's Hill College. SAVAGE, THOMAS, M.A.; Fellow of Pembroke WOOLHOUSE, W. S. B., F.R.A.S., &c.; London. College, Cambridge.

WRIGHT, Dr. S. H., M.A.; Penn Yan, New York. SCHEFFER, Professor ; Des Moines, Iowa, U.S. WRIGHT, E., B.A.; Dungannon. Soort, JOSIAH; Judge of the Ohio Supreme WRIGHT, Rev. W. J., Ph.D.; Pennsylvania. Court, Bucyrus, United States.

YOUNG, J. R.; London.

CONTENTS.

23

Solved Questions.
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. .........
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;
BẮC 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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