2 M. Falk, Method to find the gkeatest common measure 



^(•0 = f *(.0, if 



and 





These equations demonstrate evidently the truth of the theorem. 



It is of practical use to observe, that the function ^, Cr) is imme- 

 diately obtained hy omittiny in *(.t) all the termft^ which involve the m + 1 



hiijhest powers of x. 



II. If, when m < 7?, the theorem just proved is also applicable 

 to the functions *(,r) and "f'i(.r), either the greatest common measure 

 of the given functions will be found by simply applying the theo- 

 rem, or one will obtain a new function, such that the greatest com- 

 mon measure of it and of that of the functions *(./■) and ^i(.r), which 

 is of the lower degree, will be the same as the greatest common mea- 

 sure of the giA^en functions. Going on in this way applying the 

 theorem as often as possible, either the greatest common measure of 

 the given functions will be found or one must finally obtain two func- 

 tions not satisfying the conditions of the above theorem, the greatest 

 common measure of which will also be that of the given functions. 



It remains, therefore, only to shew, how to find the greatest comnidu 

 measure of two functions, neither of them involving a power of ,r as factor 

 and both of which are such, that their coefficients do not satisfy the whole 

 system of conditions (2) for the applicability of the theorem in I. Moreover, 

 allowing one of these functions, but not both, to contain a power of x as 

 factor, we may suppose them to be of the same degree, because, as by 

 supposition their greatest C(niimon measure does not involve any power 

 of ,v as factor, it will remain unaltered, if we multiply the function of 

 the lower degree by a suitable power of x. 



III. Suppose we have the functions: 



(4) . . . 



j (p{x) = «„,r" + r/,.r" ' + ... + a„_yV + a„ 

 \ 4(,r) = h,x- + 6,.r"-' -f- . . . + b„_,x + b„ 



of which both do not involve powers of x as factors and which at the same 

 time are such, that not all the coefficients in that, where the index of the last 

 coefficient that really occurs is the least, are proportional to the correspon- 



