322 PROCEEDINGS OF THE AMERICAN ACADEMY. 



while all later transformations can be taken of the type 



£ M = £y+lL rjy. = Vn+iC (43) 



After v transformations of type (40), since there can be no interchange 

 of terms among the coefficients of the different powers of the £ variables 

 in the X factor of (37), the surface will take the form 



[C + v» (v* C 2 + + P™ (Vv, 0] * (&, to = 0. (44) 



Now by the same reasoning as used for the function R in § 3, 5, if v is 

 taken large enough, the coefficients of the powers of £„ in X v will all be 

 of the type 



s = 2, 3, m. 



For any one of the functions 



there is a determinate succession of transformations of type 



Vy = £(Vn+l + <V+i) 



which will leave it of the same degree after the £ is divided out, all 

 others reducing the degree at once , i. e., if 



Vy + v (£) = Vy + <*i £ + tt 2 £ 2 + , 



we must take 



Vy = UVy+1 — "i)» 

 rjy+1 = £(Vy+2 — a 2 ), 



etc. 

 So, unless the factors 



V" + v, (Oj s = 2, 3, m 



are all equal, we must have finally some coefficient of a power of £„ with 

 the rj v present only in the E factor, and by taking y large enough we 

 come to a point where all the factors 



V' + v *(0> S = 2, 3, m, 



are equal, some of them possibly having zero exponents. 

 Then we use the transformation 



np + v.OO^n, (45) 



