326 PROCEEDINGS OF THE AMERICAN ACADEMY. 



take out the factor C, the other factor being of degree less than n r unless 

 the part (7/, £) nr has n r equal linear factors. For, if 



nr 



(V> t)n r — n (a p rj — jB pt) 

 pr=l 



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

 substitution 



V = C(vi + Si) 

 gives 



fir 

 C II (a p r/ x + dp^ — /3 p ) 

 P = l 



and leaves an absolute term in any factor for which 



a p Sj 4= /3 P , 



thus securing in the product of the factors terms of degree less than n r . 

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

 as (77, £,)n r +k- But, if all the factors of (7/, £)«,. are equal (or linearly 

 dependent) and 8j is taken so as to satisfy the condition 



a P^i = fip, p = 1» 2, « r , 



then after the factor C is divided out, we have left but one term in rj 1 nr , 

 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 rj Br , 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 

 p r (possibly to zero), except for factors taken out which are powers of 

 the r/ and £ variables. Accordingly, by a succession of transformations 

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

 p r must be reduced to the type 



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

 types 



in = t».+\yi, £m = C+1^7- ( 50 ) 



