Paper, ['Algebraical researches, containing a disquisition on Newton’s rule for the discovery of imaginary roots, and an allied rule applicable to a particular class of equations, together with a complete invariantive determination of the character of the roots of the general equation of the fifth degree, &c'] by [James Joseph] Sylvester 45 manuscript pages including one diagram showing two branches of a curve. Subject: Mathematics Published in Philosophical Transactions as 'Algebraical researches, containing a disquisition on Newton’s rule for the discovery of imaginary roots, and an allied rule applicable to a particular class of equations, together with a complete invariantive determination of the character of the roots of the general equation of the fifth degree, &c'. Received by the Royal Society on 6 April 1864. Read 7 April l864. Abstract published in Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London [later Proceedings of the Royal Society of London], Volume 13, 1864. [1864]