OF TWO RATIONAL INTEGRAL FUNCTIONS OF .T". 5- 



we have to write them thus 



a:"--{a,.T'- +... + «,) = _(a,^,.r"-'-' + . . . + a„) , 



j:"-^{b^y + • • • + '-'-) = -(6,.+i.r"-'-' + . . . + ^») , 



in order to form the equations required in Cauchy's method of elimina- 

 tion (See, for instance, M. Dostor, Elements de la Théorie des Détermi- 

 nants^ pag. 118), then to divide, one by another, these hist-written 

 equations, which gives 



«0 -^' H \-^r _ «r+,.î'""'"* 4- . . . + a 



and finally to clear of fractions and to transpose all the terms to the left 

 member. We thus obtain an equation, the left member of which evi- 

 dently is the very function /X^'), such as it is written in (6) cleared of 

 superfluous terms. 



VI. As to the method now given we make th^ following re- 

 marks, viz. 



ptiy Every time the Rule in IV is applied, it gives two new func- 

 tions, both of lower degree than the next foregoing ones, and 



2"^'^ It does not require any division, because that in V can be 

 replaced by multiplication and subtraction. 



