326 PROCEEDINGS OF THE AMERICAN ACADEMY. 



take out the factor ^"'', the other factor being of degree less than n^ unless 

 the part (17, ^)nr has n^ equal linear factors. For, if 



and not all the linear factors are equal (or linearly dependent), then the 

 substitution 



gives 



tlr 

 C n {ap-rji + ttpSi — /3p) 

 p = \ 



and leaves an absolute term in any factor for which 



apSj 4^ /3p, 



thus securing in the product of the factors terras of degree less than n^. 

 Also the degree might be lowered on account of terms in some later part 

 as (r/, Qnr+k- But, if all the factors of (77, Qn,- are equal (or linearly 

 dependent) and 81 is taken so as to satisfy the condition 



o-pOi = Pp, p := 1, 2, n,. , 



then after the factor C^ is divided out, we have left but one term in rj^''^ 

 which cannot cancel with any term from another part of the function, as 

 all later terms have as a factor some power of ^. Accordingly a suc- 

 cession of transformations of type (40), if it does not reduce the degree of 

 the part not divisible by ^, must leave a term in 77 "^ Now when the 

 reversal of type is first made, the e of (41) is zero, as is seen by con- 

 sidering the use of transformation (8) § 1, 5, Then we take out a 

 factor 7] "'■ and leave a constant term. So a succession of transformations 

 which contains reversals of type must reduce the degree of the function 

 /),. (possibly to zero), except for factors taken out which are powers of 

 the 7] and I variables. Accordingly, by a succession of transformations 

 containing a sufficiently large number of reversals of type, the coefficient 

 p^ must be reduced to the type 



9. All further transformations to be considered may be taken of the 

 types 



4 = ^f*+i ^1 '?/^ = •7m+i ^' (49) 



