B 



y researches on the method of elimination, which has first been 

 given by Bezout and developed afterwards by Cauchy, M. Cayley and 

 others, I have found the following method to obtain the greatest common 

 measure of two rational integral functions of x. 



When the greatest common measure involves a factor of the form 

 .ï'', that factor derives from the existence of factors of the form af and 

 x'' in both the given functions. Clearing the given functions of such 

 factors, the other factor in their greatest common measure will be the 

 greatest common measure of the new functions thus obtained. There- 

 fore it is evidently/ sufficient to treat the problem of finding the greatest 

 common measure of two rational integral functions, neither involving a power 

 of X as factor. 



I. Supposing we have the functions 



(1) . . . 



I <J.(,r) = .4,.^'" + A^x"^-' + ... A„,_,x + A,„, 

 \ ^(.r) = B,x" + B,x"-' + ... B„_,x + B,^ 



and m < n, we get the theorem: 



(2)- ^ = éi = = 4^, 



*(.;■) and ■i'{x) can only differ by a constant factor, if m = n;butifm<n, 

 their greatest common measure is the same as thai of the functions ^{x) and 

 '^^{x), where 



(3) -^-/.r) = J5„. + .r"-"'-^ + + B„_,x + B,, . 



For from (1) and (2) we get 



Nova Acta Reg. Soc. Sc. Ups. Ser. III. 1 



