In order to give a full discussion of the case of curves with several branches at the singular point it is necessary to discuss the type of contact that two such branches can have. This also allows us to fill in several details in our discussion of the geometry of a single branch.

We will express our result in terms of the exponent of contact of two branches. In fact, we obtain a more flexible concept by introducing the notion of pro-branch and exponents of contact of pro-branches.

The most important result is a formula relating exponent of contact to intersection multiplicity. This is the key to numerous later developments. The basic formula relates to the case when each curve has just one branch. We then develop a notation to express the type of contact of curves with several branches. It takes the form of a tree with numerical information attached, which seems best suited to describe the numerical invariants of curves with several branches.

We use our main formula to give a complete description of the semigroup of a branch. The intersection multiplicity can also be expressed in terms of the calculus of infinitely near points, and establish the essential equivalence of these two approaches, which is formalised by the notion of equisingularity.

A further section gives an application of these techniques to give a proof of a recent theorem on the decomposition of polar curves. As this is somewhat outside the main line of development of the first half of this book, it may be omitted on a first reading.