BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 325 

 If n is taken large enough, the exponent 



is made less than s, so that we have 



s > q s — ns > 0, 



or Si— I <n<Sl. 



s s 



So the term for which — is least is among those first reached in which 



s ° 



the new exponent of £ is lower than s. 



In the same way we show that, by applying a succession of transfor- 



V 

 mations of type (47), the term for which — is least is among the first 



s 



lot reached for which the new exponent of -q is less than s. But, by 



condition 2) in 5, — and — were least in the same term. So we secure 



' s s 



the surface in form (48). 



7. A further succession of quadratic transformations of type (43) as 

 applied to the surface in form (48) will reduce the singularity. This 

 follows at once by the reasoning in the first part of 5. 



D. — Succession of Quadratic Transformations in which 



THERE ARE REVERSALS OF TYPE. 



8. A succession of transformations in which there is a sufficient num- 

 ber of reversals of type will secure a surface of type (42). 



If there is but a finite number of reversals, after the last one we are in 

 the same positiou as at the start in 4, and the succession of trans- 

 formations which follows, not having any reversal of type, will enable us 

 to secure the condition derived by the method of 4. So we need here to 

 consider only the possibility of an indefinitely large number of reversals 

 of type. 



In equation (37) consider any one of the coefficients 



p r ( V ,0 = ?Pr(v,Q = tl(v,0n r + (v,On r+ l + ] 



where p r (v-> °) ^ °- 



A transformation of type (40) will give for p r a function from which we 



