BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 297 

 is analytic within the region 



\-m\<h-e t |C| < Sx4=0 3 



where h is the distance to the nearest value of ^ for which the equation 

 corresponding to 



has a critical point, e is a positive number which can be taken arbi- 

 trarily small and, having been chosen, determines an upper limit, not 

 zero, for o\. In fact, f is a continuous function of the two independ- 

 ent variables rj u £ within this region; furthermore, for any fixed value 

 of £ such that |£| < 6 1; £ a is an analytic function of r] 1 throughout the 

 region | q x \ < A — e ; and, similarly, for any fixed value of r/ x such that 

 | >7i | < h ~ e ' £ is an analytic function of £ throughout the region 



Also consider the surfaces 



in 5. Here also we have m regular points of surfaces, and as a result 

 m functions of the form 



These, by the same method of proof as above, are seen to be analytic 

 when 



III < *i-*, \v\ < **, 



where A is the nearest point in the 4-plane for which the equation 



has multiple roots for £, i. e. the smallest value of £ for which the 

 equation 



(?, 1, ?) m = 



has equal roots for $. But this is the smallest value of - for which the 



V 

 equation 



a 1, i) 



= 



* Cf. Briot et Bouquet's The'orie des fonctions elliptiques, § 28. The proof of 

 continuity there given for polynomials in two variables will apply with very 

 slight mollifications to analytic functions of any number of variables. Cf. further 

 Jordan's Cours d'analyse, I. § 206, § 258. 



