## The Collected Mathematical Papers of James Joseph Sylvester, Volume 1From 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. |

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### Contents

I | 1 |

II | 28 |

III | 33 |

IV | 36 |

V | 39 |

VI | 40 |

VII | 47 |

VIII | 54 |

XXXVI | 241 |

XXXVII | 251 |

XXXVIII | 252 |

XXXIX | 256 |

XL | 265 |

XLI | 284 |

XLII | 328 |

XLIII | 364 |

IX | 58 |

X | 59 |

XI | 61 |

XII | 66 |

XIII | 69 |

XIV | 75 |

XV | 86 |

XVI | 91 |

XVII | 103 |

XVIII | 107 |

XIX | 110 |

XX | 114 |

XXI | 119 |

XXII | 138 |

XXIII | 139 |

XXIV | 145 |

XXV | 152 |

XXVI | 155 |

XXVII | 157 |

XXVIII | 162 |

XXIX | 165 |

XXX | 181 |

XXXI | 184 |

XXXII | 198 |

XXXIII | 203 |

XXXIV | 217 |

XXXV | 219 |

XLIV | 367 |

XLV | 370 |

XLVI | 378 |

XLVII | 382 |

XLVIII | 392 |

XLIX | 396 |

L | 399 |

LI | 402 |

LII | 404 |

LIII | 411 |

LIV | 423 |

LV | 424 |

LVI | 429 |

LVII | 587 |

LVIII | 595 |

LIX | 599 |

LX | 609 |

LXI | 620 |

LXII | 627 |

LXIII | 630 |

LXIV | 634 |

LXV | 637 |

LXVI | 641 |

LXVII | 645 |

LXVIII | 647 |

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### Common terms and phrases

algebraical algebraical functions application arbitrary becomes Bezoutiant called canonical form cogredient combined commutant concomitant conic consequently constant contain continued fraction contragredient contravariants corresponding covariant cubic equation cubic function curve demonstration denominator denote derived dimensions elimination equal roots equation evectant expressed f and f factor formula given function Hence Hessian homogeneous function identical inferior limit integral function intersection invariant leading coefficient linear functions linear transformations matrix method minor determinants modulus multiplicity negative number of real number of variables obtain pairs permutations Philosophical Magazine plane positive proposition quadratic function quantities quintic function quotients real roots reciprocal relation remarkable represent respectively resultant roots of fx satisfied simple simplified residue solution square Sturm's Sturm's theorem substitutions suppose surface symmetrical synthemes syzygetic tangent theorem theory theory of elimination transformation values vanish zero