322 PROCEEDINGS OF THE AMERICAN ACADEMY. 



while all later transformations can be taken of the type 



i^ = in+lC, Vf^ = •7^+1 C (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 A' factor of (37), the surface will take the form 



[C + P-2AV^, 'C)C' + + Pm^^^^^ 0] ^(t., v., = 0. (44) 



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

 taken large enough, the coefficients of the powers of $t, in A^ will all be 

 of the type 



s = 2, 3, m. 



For any one of the functions 



>7,. + v(0 

 there is a determinate succession of transformations of type 



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

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



» 1?^ + >' (0 = ■^M + Cll ^ + ^^2 r + > 



we must take 



etc. 

 So, unless the factors 



■rii' + V, (0, s = 2, 3, m 



are all equal, Ave must have finally some coefficient of a power of ^^ with 

 the •>7„ present only in the E factor, and by taking v large enough we 

 come to a point where all the factors 



f]^ + v,(0> s == 2, 3, . . . . . m, 



are equal, some of them possibly having zero exponents. 

 Tlieu we use the transformation 



-qv + Vs (0 = Vy (45) 



