* In the present chapter I have maintained close contact with Jordan 1 . 
Another very important reference is Enriques-Chisini, vol. ii. 
| Cf. Ndther 5 , p. 267. J Cf. Appell et Goursat, pp. 184 ff. 
CHAPTER II 
DEVELOPMENT IN SERIES 
§ 1. Development of the branches at a point* 
We shall begin the present chapter by repeating a theorem 
proved in the last. 
Puiseux’s Theorem 1] The whole neighbourhood of any point 
(y) of an algebraic plane curve may be uniformly represented by 
a certain finite number of convergent developments in power series 
x i — P v yi J r a vii t v J r a vizH J r••• 
X 2 — P v y2 J !~ a v2lK J r a v22 l; v J r--' (1) 
X 3 — Pv2/3 + °W^ + a V32^v + ••• • 
Each point of the neighbourhood, except {y), will correspond to 
a single value for v and t v . 
Our present task is to find a direct way of determining such 
power-series development, without having recourse to quadratic 
transformations. After that we shall study the nature of the 
curve in the vicinity of a point, by studying the power series 
directly. 
It is to be noted that the highest common factor of all the 
exponents of t v in any one triad must be 1, otherwise there would 
not be a one-to-one correspondence between the points and the 
values of t v . The totality of points which make these series con 
vergent shall be called a ‘branch’, the point (y) the ‘origin’ 
thereof. The neighbourhood of any point is made up of a finite 
number of branches. 
The power-series developments at an ordinary singular point, 
or cusp, have already been determined in Book I, Ch. II, so if we 
know the system of transformations which carry a given point 
into an ordinary singularity, we can get all the power-series 
developments. This is the method of Nother.f We shall follow 
a modification of the classical method of Puiseux developed by 
Appell and Goursat.J 
To simplify matters, let us assume that the singular point in 
which we are interested is the origin of a Cartesian system. We