[ 



BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 303 



and all such points, a finite number in all, will be treated like the 

 critical points of the previous case. But also any point on a multiple 

 factor is a critical point of the curve, and further treatment is needed 

 for such points. 



Suppose f = a, r? = yS is a regular point of a factor <^i ($, r/) of mul- 

 tiplicity r, i. e. of the irreducible factor whose rth power is equal to 

 </>i(^) v) ^"^ "°t a point of any different factor of (ji($, -q). Then, in 

 the corresponding equation of form (7), g^ will contain a term ^'^ as the 

 lowest term in ^"^ free from rji and ^, and by Weierstrass's Theorem * 

 we can develop the function about the point in the form 



C + P^ (vi^ 0C+ + Pr (vi, Ol ^(C' vu = 0. (9) 



These functions 



P;,(vi, 0' X= 1, 2, r, 



are shown by a method similar to that used for the functions in Case I 

 to be analytic within a region 



\vi\<h-e, U|<8i, 



where h is the distance to the nearest value of rji which gives a point 

 of intersection of two different irreducible curves corresponding to factors 

 of cf) Q, rj), or to a critical point of one of the irreducible curves. 



Now none of the excepted points can be at infinity, on account of the 

 provision in 3, 3). So the points on the surfaces g = in 5 will also 

 afford developments of order (9), and by the method of Case I, we 

 have a similar region for the convergence of the coefficients of the 

 different powers of $ in the polynomial, i. e. the exterior of a circle 

 including all of the excepted points. 



Accordingly, in this case also, we represent the neighborhood of the 

 original singular point by a finite number of neighborhoods of new 

 singular points together with a finite number of functions, some of which 

 are now not analytic for the values of the arguments considered, but 

 satisfy equations of the form 



For the further treatment of these functions, we shall establish an 

 auxiliary theorem in § 2. 



* See Picard's Traite' d'analyse, Vol. II. p. 241. 



