308 PROCEEDINGS OP 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,y) = 0, « = 0, 



on the surface 



^ (x, y» 2) = 

 as required. 



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

 in the region |^| < h for which qo(y) vanishes, since the results estab- 

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

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

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

 go (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 



«^3 (^8' ^2) = ^a"* + n (2/2) ara™ -1 + + r^^iy^) = 0. (v) 



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

 reduction. Moreover, if nii = m, but 



iCs"' + ^1(^2) a^s"!-' + + r„^(yz) ^ [^3 + Pz(.y^)T^^ 



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

 that 



a^s'i + n (y2)a:3'"i-' + + ^m, (^2) = [a^s + />3 (^2)]'"', > , ,-. 



mi = m, ) 



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

 times. 



4. Suppose the function <jiz{xz^ y^) has the form (v'). Apply to the 

 surface <^3 (xg, y^, z^ = 0, (k), the transformation 



and reduce the result to the form 



^4(2^4, ^2, 22) = ^i"'^(y2) + ZzFii^i, ys. 22) = (0). (o) 



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

 than mi, 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 x^^i does 

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



