Θ THE OXFORD, CAMBRIDGE, AND DUBLIN MESSENGER OF MATHEMATICS, A JOURNAL SUPPORTED BY JUNIOR MATHEMATICAL STUDENTS OF THE EDITED BY WILLIAM ALLEN WHITWORTH, M.A., FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE; JOHN CASEY, B.A., TRINITY COLLEGE, DUBLIN ; HENRY WILLIAM CHALLIS, B.A., MERTON COLLEGE, OXFORD; JAMES MCDOWELL, M.A., F.R.A.S., PEMBROKE COLLEGE, CAMBRIDGE; AND TRINITY COLLEGE, DUBLIN ; CHARLES TAYLOR, M.A., FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE; AND WILLIAM PEVERILL TURNBULL, M.A., FELLOW OF TRINITY COLLEGE, CAMBRIDGE. "Αγγελον ἐσλὸν ἔφα τιμὰν μεγίσταν πράγματι παντὶ φέρειν· VOL. IV. C MACMILLAN AND CO. London and Cambridge. EDINBURGH: EDMONSTON & DOUGLAS. GLASGOW: JAMES MACLEHOSE. 1868. PREFACE. On the completion of the Fourth Volume, we desire to acknowledge the assistance rendered us by correspondents at home and abroad. The Volume comprises papers from France and from America; and Australia is still represented by the Chief-Justice of Queensland, who has been now for more than six years a valued writer in our pages. A greater variety than of old will be observed in the names of contributors nearer home. The assertion of our title page that the "Messenger" is supported by Junior Students, is justified by several papers by Undergraduates and by Commencing Bachelors; while the kindly interest of our Seniors is evidenced by the contributions of some distinguished authors, among whom the Sadlerian Professor of Mathematics deserves our special thanks, having kindly written out for us the solutions of all the questions in the difficult paper which he set (for the Master of Trinity) in the last "Smith's Prize" examination. As long as we enjoy the benefit of such assistance, we are sure that our readers will be satisfied with the contents of our publication. We accept with pleasure the expressions of satisfaction which we have received from various quarters as the several numbers of this Volume have been issued. At the same time we must acknowledge |