BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 319 



§ 2. Further, since the expression (f„, rj v ), n is not composed of m equal 

 factors, the part 



q (0)x a m i + q x {0)x^- l z + + ?«h(0)si"i 



from (6) which corresponds to (£„, n v ) is not composed of m equal factors, 

 and the resulting curve in (k) 



*3 CTl + niy^x^- 1 + + r mi (j/ 2 ) = 



has not m equal roots when y 2 = 0. So a single transformation of the 

 kind in § 2, 3, reduces the singularity in the neighborhood considered 

 here. 



7. The neighborhood of the original singular point is mapped upon 

 a finite number of neighborhoods of simpler points. 



At every stage the function (£ M , rj^, *£),„ contains the terms of the 

 type (£, rj) n found iu the original equation (13). So there is but one 

 singular point of the m-th order in the finite region of the 77^-plane. 

 Further, the equation (^, 1, Q m = for the value £ = cannot have 

 m equal roots since (£, v) m is not a perfect m-th. power of a linear factor. 

 Accordingly, the transformation corresponding to (8) in § 1, 4, cannot 

 produce a singular point of the m-th order. So, at each step, the 

 neighborhood of the singular point is represented by a number of regions 

 as in § 2, C, in which but one of the points of class 1) is of the mth order. 

 Further, the extra transformations (33) carry the neighborhood of the 

 singular point over into that of the new point. So, by combining all 

 the representations, as the singularity is finally reduced, we have the 

 original neighborhood mapped upon a finite number of regions as in 

 § 2, C, in which all points of class 1) are of order lower than m. 



§ 4. 



A. — The Singular Points of Special Type (continued). 



1. Case B. — The m linear factors of (f, rj, £) m are all equal. 

 The surface can be expressed in the form 



*(& v, = [F + Pn(a, Of- 2 + + Pm (r,, 0] #(6 v, 



= X(£, v ,0£&vU) = 0, (37) 



where, in X, $'" is the only term of degree m. 

 If it were iu the form 



f(u, v, w) = (ail + (3v + yw) m + (it, v, w) m+1 + = 0, 



as one of the three numbers, a, (3, y, is not zero, suppose a = 0. 



