BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 291 



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



^= l^i<S, \v\<8, |C|<8, 



which lie in the neighborhoods of" the lines 



(-^A=ol " = '■'' '' 



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 



ffAi.,vi,0 = o, <T=i,2, s, 



the coordinates being connected by the relation 



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

 (^, ■>?, 'Q which satisfy the condition 



To deal with the points for which a, /3 would be infinite, cut the 

 surface 



by the plane 



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

 are 



^^A T=l, 2, t<Tn. 



By means of a transformation corresponding to (2), 



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



^^A r=l,2, t<m, 



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 



9r (^V V>1) = 0, T = 1, 2, t<m, 



