324 PROCEEDINGS OF THE AMERICAN ACADEMY. 



reduced to a value q' r less than r — p r , in which case the sum of the 

 exponents of the three variables, 



p r + q' r + m — r, 



is less than m and reduction ensues. So it is only in the case in which 

 for every term 



p s > s, s = 2, 3, m, 



that we are not sure of reduction. Suppose the number of transforma- 

 tions after this point to be n. Then we get for the new exponent of £ 



9s + n (Ps~ *)• 

 Now by taking n large enough we can make the quotient 



n (P* - *) + 9s 



7) ™— S 



have the lowest value for the term in which — is lowest, while if 



s 



this is the same for two or more terms, we can make the fraction above 

 lowest for the one in which — is lowest. Accordingly, by a finite 

 number of transformations of type (43) we secure the condition that 



V — T V . Q 



— and so — is lowest in the same term in which — is lowest. 



r r 



6. A succession of transformations of type 



& = | M+ i£, (46) 



followed by a succession of type 



£1 = ^+117, (47) 



secures the surface with condition 5, 2) in the form 



J. (48) 



where for some particular term in X p , the rth, 



Pr <r, q r <r. j 



Consider the surface (42) with the condition 5, 2), the sth term being 



and suppose we apply to the surface n transformations of type (46), 

 dividing out each time the factor £"'. The resulting term is 



