292 PROCEEDINGS OF THE AMERICAN ACADEMY, 



the coordinates being connected by the relations 



( = vi I 



By the neighborhood of the line 



£-a T r, = 0l 



is here meant the set of points (£, 77, £) which satisfy the condition 



l^-u^l^hl, |fl<«|*|, \v\<8. 



The singularities of the surfaces 



9A^ Vi, O = o, 



ff T it* v, = 



at the points (0, 0, 0) are at most of order m. Their further proper- 

 ties will be considered later. 



Let G be an arbitrarily chosen (large) positive quantity, 8 a second 

 suitably chosen positive quantity : then any point of T, for which 



I*i<g|ci, \v\<e\t\, o<m<s, 



is carried by the transformation (4) into one of the neighborhoods above 

 considered on the surfaces g a = 0. If £ = 0, but £, 77 do not both van- 

 ish, then the point (f, 71, £) is carried by (5) into one of the neighbor- 

 hoods considered on the surfaces g T = 0. 



In (3), the function <j> ($, rj) contains the term $ m by 3, 2). Apply the 

 transformation 



V-P= *7i, (6) 



whence (3) takes the form 



* (6 v, = C" Oi (I vi) + t Xi (?» ft> 0]. 



In (j> x (£, tij), take out all terms not containing rj u so that 



_ « _ ^a _ 



01 (£ *h) = n (£ — a„) + 77 1 i^ (£, ??!), (U! + + ^ = m. 



<r = l 



Then make the transformation 



* ~ «„ = C (60 



and we have 



* (6 ,, = r tc n' & + a a - v )' v + 71 * (* a , in) + : x (k %. 0] 



<r 1 =: 1 



= Vg 9 (*„ % = (7) 



