BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 329 



type (46) or (47) can be considered zero, while in the other case we 

 have exactly the initial conditions of 6, the result of which then can be 

 secured in any case whatever. 



12. In the case of surface (48) any succession of transformations of 

 types (49) and (50) will finally reduce the degree of the singular point. 



Consider the term 



m — r 



Any transformation of type (49) adds to the exponent of ^, jo,. — r, and 

 as pr < r, the exponent of ^ is reduced. In the same way we see that 

 any transformation of type (50) reduces the exponent of the rj variable. 

 So in any case, by virtue of the reduction of degree, we must have finally 

 either 



p,<r — q^ or q, < r ~ p,, 



in either of which cases the sum of the exponents of the three variables 



{jn — r) + p^-\- q, 



is less than m, and we have reduction of the singularity. 



§ 5. 



Parametric Representation of the Neighborhood of the 

 Original Singular Point. 



"We have shown that in all cases T, the neighborhood of a singular 



point, can be mapped upon a finite number of regions t^, t^, <„ as 



defined in § 2, C. Apply a properly chosen transformation to each point 

 of class 1) and repeat the operation on each set of resulting points of the 

 same class as they are formed. We have proved also that after a finite 

 number of operations all the resulting points of class 1) are of order 

 lower than m. Then, by continuing the process, it follows that, after a 

 finite number of transformations, all points of class 1) must disappear, 

 and so we shall have left only regions of class 2). Each of these regions 

 ailmits of representation by means of a finite number of sets of para- 

 metric formulae of type {A). 



Classify all the singular points which present themselves in groups as 

 follows : — 



In the first group, place the original point ; in the second, all singular 

 points derived from it by the first quadratic transformation, together with 

 whatever auxiliary transformations accompany it ; these points corre- 

 spond to the singular points of the curves that represent the irreducible 



