308 PROCEEDINGS OF THE AMERICAN ACADEMY. 



secure a limit for the values of x and 2, and thus we have represented 

 a neighborhood of the curve 



<f>(x, 20 = 0, « = 0, 



on the surface 



$ (a:, y, z) = 

 as required. 



Now, however small the neighborhood we shut off about the points 

 in the region \y\ < h for which q (y) vanishes, since the results estab- 

 lished above would hold also in a circle of radius h x > h, but still less 

 than the radius of convergence of the series for p (y) in (/3), we can fill 

 up the remainder of the circle of radius h with circles within which 

 g (y) does not vanish, these circles overlapping at all points the bounda- 

 ries of the excepted neighborhoods and not reaching up to the excepted 

 points. Within each of these circles we have a development of type 

 (k). Consider one of these new circles. We want to consider the 

 neighborhood of the curve 



& (*» ft) = *3 m i + r x (y 2 ) x z m - 1 + + r mi (y 2 ) = 0. (v) 



If this is a multiple curve of the m x -th. order and m x < m, we have 

 reduction. Moreover, if m x = m, but 



«."» + r x (y 2 )x z m ^ + + n„,(y 2 ) 4= [> a + /> 3 (y 2 )]"\, 



we also have reduction. We need consider, therefore, only the case 

 that 



*3 m ' + rx (j^W^ 1 + + r mi (y a ) = [x, + PsCya)]" 1 !, > , ,. 



m x = m, > 



and show that this case can repeat itself at most but a finite number of 

 times. 



4. Suppose the function <£ 3 (x 3 , y 2 ) has the form (v'). Apply to the 

 surface <J> 3 (x s , y 2 , z 2 ) = 0, (k), the transformation 



x s + p 3 (yz) = x i> 



and reduce the result to the form 



^O^ y» z*) = x^Efa) + 22-^4(^4) yt, 22) = (0). (o) 



If any term in z 2 F i (x i , y 2 , z 2 ) is of degree in x i and z 2 together less 

 than m u it appears at once that we have a line of lower order. So we 

 assume there are no such terms. Also, as the coefficient of a; 4 m i does 

 not vanish identically in y 2 (in fact, not at all) no transformation of 



