BLACK. THE NEIGHBORHOOD OP A SINGULAR POINT. 289 



The surface $ = corresponds in the neighborhood considered, point 

 for point, to the surface F = 0, and thus it is only necessary to prove 

 the theorem for <t> = 0. 



We may assume that of the irreducible factors* of 4> there are none 

 of degree lower than m vanishing at the point (0, 0, 0), for otherwise 

 each of such factors could be treated separately by the methods here 

 used, and the results combined. This provision excludes the case in 

 which one of the variables has equal roots for all values of the other 

 two in the neighborhood of the point (0, 0, 0). 



4. The quadratic transformation 



*=?: v = v (2) 



reduces <i> (ft r;, £) to the form 



<Kft^0 = ^W>(ft^) + £x(fti0] 



= £ m <b(lv,Q (3) 



where, au arbitrarily large positive number r having been chosen at 

 pleasure, 8 can be so determined that the function <i> will be analytic 

 when 



|?|<r, \v\<t, |C|<«. 



Equation (3) follows at once from the intermediate form 



4> a, v , o = c id v, i)<» + c(#, t. iu + £ 2 (i, v, i)»+2 + ]• 



We now proceed to the proof that the function (£, 77, £) is analytic 

 within the above limits. 



Let <P (ft 17, = 2 J . x . £ rf £* , t + / + * > m , 



and suppose it to be convergent when 



|f I < h, \ v \ < /,, |C| < h h> o\. 



Then, for the general term, we have 



\A ijk \^+ k <M, 



M being a positive constant. 

 By transformation (2) 



<*> (ft V , = 2 j* ? 7 r i+ * 



* For the definition and the fundamental properties of the irreducihle factors 

 of an analytic function of several variables, which vanishes in a point, cf. Encyclo- 

 piidie der mathematischen Wissenschaften, II. B. 1, Nr. 45. 

 vol. xxxvil. — 19 



