From the Preface: ""The longest paper in volume 1 is ""On the Theory of the Syzygetic Relations of two Rational Integral Functions, comprising an application to the Theory of Sturm's Functions"", and to this many of the shorter papers in the volume are contributory ...the volume contains also Sylvester's dialytic method of elimination, his Essay on Canonical Forms, and early investigations in the theory of Invariants. It also contains celebrated theorems as to Determinants and investigations as to the Transformation of Quadratic Forms and the recognition of the Invariant factors of a matrix.