3o0 PROCEEDINGS OF THE AMERICAN ACADEMY. 



factors of (jt {i, rj), to the points of intersection of two such curves, and 

 to the points of class 1) in § 2, 1. In the third group place all singular 

 points deriv^ed in a similar way from those of the second group, etc. 



Suppose 71 to be the number of the last group in which there are 

 singular points. From what we have proved, n must be finite. 



The neighborhood of a point in the «th group is represented by the 

 neighborhoods of a finite number of regular points, together with a finite 

 number of regular regions, and so by a finite number of parametric 

 formulae of type (A). The neighborhood of a point in the (n — l)st 

 group is represented by the neighborhoods of a finite number of points of 

 the wth group, together with a finite number of regular regions, however 

 small the neighborhoods of the singular points are taken ; but as the 

 neighborhood of any point in the nth group is represented by a finite 

 number of parametric formulae of type (A), the same follows for any 

 point of the (71 — l)st group, using the intermediate transformation to 

 get the parametric formulae. 



This reasoning can be carried on until the original singular point is 

 reached, since the mapping of the neighborhood of the original point 

 upon a finite number of regions of classes 1) and 2) applies to each of the 

 later singular points also, and then furnishes the step by which we know 

 that we can always pass from the {v + l)st to the vth group. 



Thus we have the coordinates $, -q, ^ of the surface 



expressed in parametric formulae of the desired type, the parameters 

 being in general coordinates of points of some simple surface. Then by 

 using the intermediate transformations connecting x, y, 2 with $, -q, ^, we 

 represent the first set of coordinates in the desired form. 



