282 PROCEEDINGS OP THE AMERICAN ACADEMY. 



and the neighborhood of the original point is represented by the neigh- 

 borhood of the curve 



<^(r, cr)=0, C=0, (c) 



on the surface 



^ (t, (T, = 0. 



3) The neighborhood of the curve (c) is included in the domains of 

 a finite number of points which are 



a. regular points of the curve (c), the domain of each being repre- 

 sented by a single jiower series 



T=p(cr,0; (d) 



h. critical points of the curve (c), the domain of each being repre- 

 sented by an equation of the form 



^ + pi(cr, r^-^ + + ^^-l(cr, ^ + p^ (cr, 0=0; (e) 



c. points at an infinite distance on the curve (c), the domain of each 

 being represented by an equation of form (d) or (e) in the variables 

 T], o"!, 1], where 



T 



1 



T 



1 ' 



<j cr 



O"! 5 i(r = 7] 



4) The selection of the points in 3) depends upon the character of 

 the curve 



c^ (r, cr) = . 



a. If is irreducible, all points of class 3) b are first taken, then all 

 points of class 3) c, these being regular; finally a finite nnmber of 

 points of class 3) a. Here, all the points selected, if singular, are of 

 order less than 7n. 



/). If <^ is reducible, but contains no multiple factors, the same selec- 

 tion of points holds as in a, but there may occur a singular point of 

 order m. 



c. If <^ contains multiple factors, all critical points of the curves cor- 

 responding to any factor, together with all points of intersection of two 

 different factors, are first taken, then all points of class 3) c, these being 

 possibly singular ; finally, a finite number of regular points of the several 

 curves corresponding to the different factors of <^, these last points being 

 possibly singular points of the surface. In this case, there may occur 

 a number of singular points of order m. 



2. Treatment of Points Determined in 1. The same treatment as in 

 1 is apjilied to each of these points and to each of the corresponding 



