xxii CONTENTS 
§ 3. Reduction of singularities 
Necessary and sufficient conditions for the reduction to a curve 
with only double points, or to a non-singular curve . . 399 
§ 4. Reduction to linear systems of minimum order 
Sufficient conditions for reduction of order .... 401 
§ 5. Reduction of curves of low genus 
Reduction of linear systems of rational curves .... 403 
Reduction of elliptic curves and systems of such curves . . 404 
Reduction of curves lacking adjoint systems of high index . . 406 
CHAPTER III 
TERNARY APOLARITY 
§ 1. Linear systems and hyperspace 
Passage to the general linear system , . . . .410 
Representation in hyperspace . . . . . . .410 
§ 2. Apolarity 
Formation of an invariant operator . . . . . .412 
Definition of apolar forms . . . . . . .412 
§ 3. Apolarity between forms of different orders 
Extension of apolarity, polar curves . . . . .413 
§ 4, Expression of forms in terms of perfect nth powers 
General theorems about such expressions . . . . .415 
Particular types . . . . . • . . .416 
CHAPTER IV 
SPECIAL CURVES IN LINEAR SYSTEMS 
§ 1. The pencil 
Invariant number for a pencil . . . . . .417 
The inflexional locus ........ 420 
§ 2. Two-parameter nets 
Invariant numbers ........ 422 
The Jacobian ......... 423 
§ 3. The Laguerre net 
Definition of the Laguerre net 423 
Properties of the Jacobian ....... 424 
NO! 
§ 1. General form 
General expressior 
Invariant number 
Curves cut from a 
§ 2. Singular poh 
Various arrangerai 
Analytic expressio 
Determination of i 
§ 3. The inflexion 
Determination of 
§ 4. Projective the 
Properties of the i 
Number of curves 
§ 5. Systems depe 
Invariants for a t\ 
Number of curves i 
curve 
Number of curves 
THE GEIS 
§ 1. Fundamental 
Definitions and ge: 
Fundamental poin 
§ 2. N other's facte 
Factorization of tl 
Genesis of the firm 
§ 3. Applications 
Fundamental poin 
Montesano’s theor 
§ 4. The identities 
Distribution of the 
Coordinated group 
Properties of unsy