BLACi;, — THE NEIGHBORHOOD OF A SINGULAR POINT. 297 



is analytic within the region 



\vi\<h-e, |^|< h^^O, 



where h is the distance to the nearest value of tji 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 81. In fact, ^ is a continuous function of the two independ- 

 ent variables t^^, ^ within this region ; furthermore, for any fixed value 

 of t, such that | ^ | < Sj, f is an analytic function of 771 throughout the 

 region | ?;i | < A — e ; and, similarly, for any fixed value of 771 such that 

 I 'Ji I < ^ — ^) f 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 



\1\< h^-e,, \rj\ <8^, 



where k is the nearest point in the {-plane for which the equation 



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

 equation 



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

 equation 



(W-) =0 



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

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

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

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



