BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 285 



has a singular point at 



u = 1, V = 1. 



So the first transformation is 



i; = (o- + 1) £ V (1) 



which, applied to (k), gives 



£ 2 (r 2 - o- 2 + to-0 = (m) 



and 



x (t.,«,)=£ 2 o- 2 (£ 2 +4). (n) 



Now the set of transformations to which Kobb is naturally led in this 

 case is the following : — * 



T = Ti & O" = 0^ & £ = £j 



*"l = T 2 £ 2 0"l = 0"2 £2 £l — £2 



whence 



T = TV C, O- = OY £/> £ = £r- 



But this substitution in (11) gives 



x (v, w) = V r+2 <r* (C 2 + 4) 



in which the exponent of £ r increases indefinitely with r. 

 6. In the case in which the curve 



c£ (r, «x) = 



has multiple factors, the regular points of such factors taken in 1, 4) c 

 are possibly singular points of the surface, whose domains are repre- 

 sented by equations of form (e). When a further quadratic transfor- 

 mation is applied to such a point, we are not warranted in assuming 

 that the resulting developments will represent the whole of the domain 

 of the point considered. f Kobb makes this assumption in proposing to 



* This set, combined with the transformation (1), possesses all the properties 

 required by Kobb in (g), (h), and (i) ; its appearance here invalidates his proof. 

 It can easily be shown, moreover, that the most general set of transformations 

 which he could use in this case would produce the same condition as shown here. 



t The development about the point first considered, to begin with, is a relation 

 im Kleinen ; it becomes, however, on passing to the later transformations, a relation 



