Inferior Limits to the real Roots of any Algebraical Equation. 139 



only variations of sign ; e. g. if the entire series be q l} q q , q 3 , q 4 , and 



the corresponding algebraical signs are H h , we shall have 



the two sequences q x >q<£ q-s>94- If the entire sevieshe q v q 2 ,q 3 ..,q 15 , 



and the signs be b ^ — h + + H h + -f H — , then the 



sequences to be taken will be 



and so in general. 



Suppose, now, that q P +i, q p+2 j • • • q p+i are the terms of any 

 one such sequence. Then, provided that 



aild ^ + ^^ > 



(it being understood that the values of fi v fi^ . . . //, t _i are per- 

 fectly arbitrary, except being subject to the condition of being 

 all positive, and that there are as many distinct and independent 

 systems of such values as there are sequences of variations of 

 sign), it will continue to be true (and capable of being demon- 

 strated to be so by precisely the same reasoning as was applied 

 to the demonstration of the lemma in its original form) that the 



denominator of ... — will have the same sign as the 



01+ ?2+ q r 

 product q\-q<2>q%* . • q r * It will be observed that, as regards 

 the residual quotients not comprised in any sequence, their values 

 are absolutely unaffected by any condition whatever. As a 

 direct consequence from this lemma, we derive the following 

 greatly improved Theorem for the discovery of the limits. 



Let, as before, fa =0 be any given algebraical equation; <\>{x) 

 any assumed arbitrary function of (#) of an inferior degree to 

 that offa; and let 



££__L 1 1_ J_ 



/*-x 1+ x 2 + x 3 +-"x r ; 



and let the leading coefficientsof X X ,X 2 , X 3 ,...X r heq v q 2} q 3> ... q r) 

 and let this latter series be divided into sequences of variations 

 and residual terms not comprised in any such sequence, as ex- 

 plained above. Let the X's corresponding to the residual terms 

 be called V V V 



and let the successive sets of % 8 corresponding to the sequences 

 be called respectively 



v„ v 2) ...v„ 



V'„ V' 2) ...V' p , 



V„ v" 2 ,...vy 



(^(^...(VJ. 



