x PREFACE 
With regard to the choice of method, preference is given, in 
the first instance, to algebraic procedure. Large portions of the 
work are written according to the spirit and methods of the 
Italian geometers, to whom, indeed, the whole is dedicated. 
It would be quite impossible to describe the extent of the 
writer’s obligation to them. Yet behind the Italians there 
stands one whose contributions are even greater, Max Nôther. 
Besides algebraic methods, there is much use of geometric ones, 
especially those involving the projective geometry of hyperspace. 
Transcendental analysis takes a secondary place, but has been 
treated at least as an honoured guest in the house. It is assumed 
that the reader will not have heart failure at the mention of 
a Riemann surface; in studying the fundamentally important 
topic of linear series of point-groups on a curve, the relation to 
Abelian integrals, and Abel’s theorem is insisted on. The Chasles- 
Cayley-Brill correspondence formula recurs again and again, 
but no one has yet been able to prove all the connected theories 
without the help of Abel’s theorem and the use of theta func 
tions. The needed properties of these latter are given without 
proof. The only distinct methods which have never been used 
are those of the theory of algebraic numbers. The reader who 
wishes to study the properties of plane curves from this point 
of view is referred to the solid work of Hensel and Landsberg, 
Theorie der algebraischen Funktionen. It would be hypocrisy to 
attempt to give a sound justification for all the choices made. 
Every writer must reconcile, as best he may, the conflicting 
claims of consistency and variety, of rigour in detail and elegance 
in the whole. The present author humbly confesses that, to him, 
geometry is nothing at all, if not a branch of art, and the under 
lying force which compels him to treat any particular topic, or 
to handle it in any particular way, is either that he is ignorant 
of any other, or else that his aesthetic sense dictates the choice : 
it pleaseth him so to do. 
J. L. C.