OF TWO RATIONAL INTEGRAL FUNCTIONS OF a: 3 



ding coefficients in the other; and let the functions /(a;), /^(j;), ..., /;_j(^) 

 be formed accordingl}' to the law expressed by the equations: 



• f{.r) = a„4(,0-6„(p(,.), 



(5) • • • /; (•'•) = («0.^- + «:■'■ + <J 4 (•'•) - (b,yi- + b^.v + 6,) cp (,r) , 



For these functions the following laws are found to hold, viz. 



1) The degree of everi/ function /(x) , . . . , fn-xix) can never sur- 

 ])»■■<!< n — 1. 



In fact we have (0 < r < n—l) 



/X-r) = C'^,-'''' + . . + '/,.) [.r"-'-(6,.r'- + . . + A,.) + A,+,r"-^-' + . . + 6„] — 



— (b,,v'- + .. + hr) br'-'{a,,v'- + .. + «,) + «,+>.r"-'-' + .. + «„] 

 or 



(6) /X-'O = (%'V'+- ■ +«.) (^,.+f'-"-'-' +• • l>,:)—(J>„x'+. . + b,) (a,.^,v'-^- ' + . . +a„). 



2) From the equations (5) it is immediately seen, that everi/ com- 

 mon measure of Ç^^x) and 40'') divides all the functions /(.«) , . . . , /'„_i(.r). 



3) When between the coefficients in Cp^,;-) and 4G'') '^'^''^ exist the relations 



^^ \,'\ ^ 0- <70, 



///(' function /'(.(■) « involved as factor in every one of the functions f^{x') ^ . . . , frQv). 

 For, if ^ < ?• , we have in virtue of (7) 



or 



