BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 291 



Now, the transformation (2) being made, the points of the region 



T: |*| < 8, h|< 8, |C|<8, 



which lie in the neighborhoods of the lines 



> cr = 1, 2, 5, 



can, with the exception of the point (0, 0, 0), be transformed in a one- 

 to-one manner on the neighborhoods of the points (0, 0, 0) of a set of 

 surfaces 



9« (£r> VU = ° . Or = 1, 2, S , 



the coordinates being connected by the relation 



r ( 4 ) 



*=£foi + /?> S 



By the neighborhood of the above line is here meant the set of points 

 (£, 7/, £) which satisfy the condition 



|*-«**|<«|C|, h-£C|:S«|C|, ICK8- 



To deal with the points for which a, /? would be infinite, cut the 

 surface 



4> (£ r/, = 

 by the plane 



C = o. 



The equations of the tangents to the curve of intersection at (0, 0, 0) 

 are 



f- t=1, 2, «<m. 



By means of a transformation corresponding to (2), 



the points of T which lie in the neighborhoods of the lines 



£ - « T 77 = ) 



[ T= 1, 2, < < TO, 



£ = oj 



can, with the exception of the point (0, 0, 0), be transformed in a one- 

 to-one manner on the neighborhoods of the points (0, 0, 0) of the set 

 of surfaces 



SV(l T ,^£) = 0, t= 1, 2, *<m, 



