4 M. Falk, Method to find the greatest common measure 

 or finally 



/;(.ï) = 1 (a^a;'- + . . + a^^fXœ) . 



4) When the conditions (7) exist and the expression 

 a^b,+, — ar+J>o 



does not vanish^ every common measure of f{x) and /r+i(.?') divides both the 

 given functions £p(,c) and ^C*)- 



For in this case we get without difficulty 



{a,h,^—a^^,h,)(p{x) = («o^'^""' + • • + «.+.)/C'0 — «O.À+.C'-) . 

 (a„6,+, — a„+,6o)4(.r) = {b,x'^' + . . + b ,.^,)f{x) — bj,^^,{x) . 



IV. The foregoing results may be summed up in the following 

 practical 



Rule. // a^h^ — a^b^ does not vanish, construct the functions f{x) 

 and f^Qc); if, on the contrary, 



ajjf. — a^b^ = 



for all /*<'■•) but 



does not vanish, construct f{x) and fr+,{x). In each case the greatest common 

 measure of the tivo functions thus constructed will also be that of the given 

 functions. 



If the two given functions have a common measure involving x, 

 the repeated application of this rule and of the theorem in I must ne- 

 cessarily at last lead to two functions only differing by a constant factor, 

 and thus the greatest common measure of the two given functions is 

 evidently found. 



V. In order to facilitate the application of the method, we make 

 the following remark as to the construction of the functions /.(.c). 



Putting 



4Gr) = 0, 



