Paper, 'On a theory of the conjugate relations of two rational integral functions comprising an application to the theory of Sturm's functions and that of the greatest algebraical common measure' by J J [James Joseph] Sylvester A supplement to this paper is found at PT/47/8. Subject: Mathematics Published in Philosophical Transactions as 'On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure'. Received and read by the Royal Society on 16 June 1853. 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 6, 1854. [1853]