## Principles of Geometry, Volume 5Henry Frederick Baker (1866-1956) was a renowned British mathematician specialising in algebraic geometry. He was elected a Fellow of the Royal Society in 1898 and appointed the Lowndean Professor of Astronomy and Geometry in the University of Cambridge in 1914. First published between 1922 and 1925, the six-volume Principles of Geometry was a synthesis of Baker's lecture series on geometry and was the first British work on geometry to use axiomatic methods without the use of co-ordinates. The first four volumes describe the projective geometry of space of between two and five dimensions, with the last two volumes reflecting Baker's later research interests in the birational theory of surfaces. The work as a whole provides a detailed insight into the geometry which was developing at the time of publication. This, the fifth volume, describes the birational geometry of curves. |

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### Contents

INTRODUCTORY ACCOUNT | 1 |

Greatest possible number of double points of a plane curve 1011 | 10 |

Examples of elliptic curves Coresiduation Salmons | 18 |

THE ELIMINATION OF THE MULTIPLE | 24 |

Examples of transformation of curves 3134 | 31 |

The parametric expression of a branch of a curve 3943 | 39 |

General theorem for infinities of a rational function 4649 | 46 |

Examples of Abels theorem 5557 | 55 |

INTEGRALS RELATIONS AMONG PERIODS | 136 |

Interchange of argument and parameter Converse of Abels | 144 |

Important properties of the fundamental integral functions 151155 | 151 |

The functions reciprocal to the fundamental integral func | 157 |

The structure of a certain fundamental rational function 165169 | 165 |

Explicit form of relation for interchange of argument | 171 |

Some particular applications Return to theory of special | 177 |

Interpretation of the formulae as illustrating general prin | 192 |

MENTALS OF THE THEORY OF LINEAR SERIES | 59 |

Equivalent or coresidual sets of points on the curve | 65 |

Applications of the RiemannRoch formula Cliffords | 78 |

The existence of a rational function with assigned poles | 86 |

The theory of special sets an extension of Cliffords theorem | 94 |

Related theorem as to intersections of plane curves 105107 | 105 |

THE PERIODS OF ALGEBRAIC INTEGRALS | 111 |

Riemann surfaces 121124 | 121 |

Curve systems on a surface The EulerPoincare invariant 132133 | 132 |

Curves which are the complete intersection of two surfaces 201208 | 201 |

Curves which are the partial intersection of two surfaces 208209 | 208 |

Examples Curves not determinable by three surfaces | 209 |

Examples Composite curve intersection of two surfaces | 216 |

Another proof of the determination of the canonical series | 226 |

The greatest genus possible for a curve of given order | 234 |

243 | |

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### Common terms and phrases

adjoint curves aggregate algebraic arbitrary arbitrary points arises assumed boundary branch called circuit closed coefficients coincident columns common complete complete intersection consider constants contain Conversely coordinates corresponding cubic curve curve of order defined definite denote depends determined dimensions double points equal equation exists expression fact follows formula freedom fundamental further genus give given Hence homogeneous independent infinite infinities integral intersections involve linear series linearly lying meet multiple points namely neighbourhood obtained original parameter particular pass places plane curve poles polynomial positive possible power series prime projection proved quadric surface quartic quintic curve rational function reduced regarded remaining remarked represented respectively result rows satisfying shewn simple single space suppose surface surfaces of order symbols taken tangent line theorem theory transformation values vanishes variable zero