BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 283 



resulting points in turn, so long as they are singular. If after a finite 

 number of such processes, all the resulting points are regular, then by 

 combining the results it is assumed that the neighborhood of the origi- 

 nal point is represented by the domains of a finite number of regular 

 points, and so by a finite number of parametric formulae as desired. 



3. Proof that a Finite Number of the Processes of 1 will be Sufficient 

 to make all Points in 2 Regular. Starting with the surface 



f{u,v,w)=^0, (f) 



in which the singular point considered is at the origin, the transfor- 

 mations in 1, 1) and 2) are combined in the form 



M = (ai T + /8i o- -f yi) n 



V = (aaT -f ^2 o- + yo) C J- (g) 



w= (osT + P^a- + ys)^^ 

 We can assume that 



72 + , ys ^ 



by making, if necessary, upon f (u, v, w) a suitable homogeneous 

 linear transformation. Then the next set of transformations, in 2, can 

 be expressed in the form 



r=(ai'Ti-f ^I'cTi + yiOCi) 



o-=(ao'Ti +/3o'(ri -f y.OCi \ (h) 



in which ys' -i^ 0,* and the later sets of transformations are of the same 

 type with the corresponding ys's : 



y3"t0, y3"'t0, etc. 



So we consider a succession of transformations of type (g), which give 

 a succession of surfaces with multiple points each of order jn. These 

 transformations will combine in the form 



w = [ys ys' ys" ys"' + (t,., cr„ I) ] ^. = [?« + (r,, o-„ 4) ] L ) 



where the symbol (t,., o-^, C) represents in the expression in which it 

 occurs all of the variable terms, and r2 =|= 0, Fg 4= 0. 



* To secure this, Kobb makes unwarranted use of a quadratic transformation, 

 which, however, might be replaced by a homogeneous linear transformation. He 

 also overlooks one class of transformations which will arise. (see 4). 



