BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 305 



on a finite number of regions r u t 2 , t , which fall into two 



categories : — 



1) the region r t (i = 1, 2, k) is the neighborhood of a singular 



point of order < m ; 



2) each of the neighborhoods of 1) having been determined arbitrarily 



small, the region t s (i = k + 1 , v) is then a regular piece of an 



analytic surface, represented in its whole extent by a single set of para- 

 metric formulae of the type {A). 



By the neighborhood of the curve 



x+p(y) = 0, s = 0, 



is meant the set of points (cc, y, z) satisfying the relation 



\y\<h, |*| < 8, \x + p(y)\<e. 



B. — Proof of the Lemma. 



2. To prove the lemma we begin by expressing equation (a) by means 

 of (J3) in the form 



* (*, y,z) = [x+p (y)y» E(x, y) + z*(x, y, z) = 0, (y) 



and then making the transformation 



x + p (y) = x x , (S) 



thus obtaining the equation 



$0, y, z) = 4>j (a?!, y, z) = x 1 m E(x u y) + zip x (x u y, z) = 0. (y') 



Here, the function E (x u y) is analytic and different from zero in the 

 neighborhood of any point x x = 0, y = y , (\y | < h), which corresponds 

 to the neighborhood of the point x = p (y ), y , and lience E (x x , y) is 

 analytic throughout a region including in its interior the region 



l*i I < e > \y\ < h > 



if the positive quantity € is suitably chosen. A similar remark ap- 

 plies to the analytic character of the function \p x (x u y, z), and hence 

 <!>! (x u y, z) is an analytic function of its three arguments throughout 

 a region including in its interior the region 



l^i I < e > |y| < A » 1*1 < s - 



Now express equation (y') in the form 



*i(*n y ? z ) = 2p,..(y)^i r 2 s + F(x 1} y, z) = 0, (e) 



where 



< r + s = mi < m, 

 vol. xxxvii. — 20 



