310 PROCEEDINGS OF THE AMERICAN ACADEMY, 



where 



9 F 

 Substitute in equation (v) for x from (p), using for F and ^r— their 



dx 



ation (v) for x iroi 



values as derived above, and we have 



9F 

 ;j-' L, {x,, y, z) F, + z^ra-x^M, (x,, y,z)^^— = z^ i?i {y, z). 



The left side of this equation is divisible by 2"'"''"'' and so the right side 



must be also. 



^(m — 1) ^ X, 



and we have an upper limit for v, the number of transformations which 

 leave the order of the multiple line unchanged. 



The securing of the regions of class 2) in 1, follows from the reduc- 

 tion just proved. If for all multiple curves of order n or less the lemma 

 is assumed to hold, this reduction establishes it for all curves of order 

 w + 1, since by it the neighborhoods are represented by those of lower 

 order. But we know it to be true for curves of the first order, and so 

 by mathematical induction we establish it for curves of all orders. 



5. Tlie neighborhoods of singular points in 3, if they are of the m-th 

 order can be taken along the curve 



4>{x,y) = 0, 2 = 0, 



on the surface 



^{x,y,z) = 0. 



In fact, the first lot of points excepted, those for which in equation 

 (yi) 9o (y) vanishes, are along the line 



X2 = 0, 22 = 0, 



which is connected with the original curve by the one-to-one transfor- 

 mations (8) and (^). Also so long as the multiple curve does not break 

 up into simpler curves, the neighborhoods corresjiond, and when this 

 reduction takes place we can cut out the neighborhoods of the points 

 common to all of the resulting curves by cutting out neighborhoods 

 along the original curve for the same values of y. 



C. — The Reduction of the Original Singularity. 



The transformations hitherto considered, when applied to the original 

 surface ^ (^, r], C) = 0, make it possible to map the neighborhood of 

 the point (0, 0, 0) of that surface on a finite number of regions which 

 are of two classes : — 



