292 PROCEEDINGS OF THE AMERICAN ACADEMY, 



the coordinates being connected by the relations 



By the neighborhood of the line 



is here meant the set of points (f, -q, C) which satisfy the condition 

 The singularities of the surfaces 



ffr (^^' V, 0=0 



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 j^ositive quantity : then any point of T, for which 



\^\<GU\, \v\<G\C\, 0<\i\<8, 



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

 considered on the surfaces y^ = 0. If ^ = 0, but ^, r] do not both van- 

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

 hoods considered on the surfaces g^ = 0. 



In (3), the function <^ ($, r}) contains the term f "* by 3, 2). Apply the 

 transformation 



V-^^Vi. (6) 



whence (3) takes the form 



^ a, V, = r [<^i (?, vi) + (xi (I vi^ 0]. 



In 4>i (t? Vi)f take out all terms not containing rji, so that 



i>i (i, Vi) = n (^ — a^) -f 7/1 1//1 (^, Tji), fii + + ft, = m. 



Then make the transformation 



^-a. = 4 (6') 



and we have 



4) {$, ^, = r [t/ i' (4 + a, - a^.T"' + V^^ a., Vi)+Cx (i.^ Vi^ 0] 

 (r';= 1 



= ^"'ff. (I., Vu = (7) 



