BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 305 



on a finite number of regions ti, T2, t , which fall into two 



categories : — 



1) the region Ti(i ■= \, 2, k) is the neighborhood of a singular 



point of order < m ; 



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



small, the region tj ( i ^= k -\- 1 , v) is then a regidar 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, 2 = 0, 



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



\y\<h, \z\<h, \x + p{y)\<^. 



B. — Proof of the Lemma. 



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

 of ifi) in the form 



«D {x, y,z) = lx+p (^)]"' E(x, y) + z^{x,y,z)z^O, (y) 



and then making the transformation 



x-\- p{y)=x^, (S) 



thus obtaining the equation 



$(a;, y, z) = ^^(x^, y, z) = x{'E{x^, y) + z^^{xi, y, z) = 0. (y') 



Here, the function E{xi, y) is analytic and different from zero in the 

 neighborhood of any point Xi ^ 0, y = yQ, Qy^ | < h), which corresponds 

 to the neighborhood of the point x^ = ;; (yo)i y^^ and hence E (xi, y) is 

 analytic throughout a region including in its interior the region 



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

 plies to the analytic character of the function ij/i (xi, y, 2), and hence 

 $1 (xi, y, 2) is an analytic function of its three arguments throughout 

 a region including in its interior the region 



l^il < e> \y\< h, |^|<s. 



Now express equation (y') in the form 



"^1 (a^n y, «) = 2p,. (y) x{!? + F{x^, y,z) = Q^ («) 



where 



< ?• -|- s = mi < »j, 



VOL. XXXVII. — 20 



