-^^ '^H^. of Btmm's Theorem.^ r :. . .-., 44^ 



expand '^-J^, [f[x)hemQ of the ntli degree in (i),] algebraically 



under the form of a continued fraction 

 1 



i^" 



td^ ^UK 



gmiaJ ilro. . 



^i'nir:\'hm. ... -. Q^' 



where Qj, Q^, Qg, . '. .Qn niay be supposed linear functions of a? 

 (although, in fact, this restriction, as will be hereafter noticed, is 

 unnecessary), the denominators of the reverse convergents 



l_ Qn-i Q»-i.Q„-2...Qi-&c. 



1' Qn' Q«.Qn-i-r ••• Q«.Q«-i...Qi-&c. 



will be signaletically equivalent with the Sturmian series of 

 functions for determining. the. number of real roots of fx within 

 given limits; in fact, ,, ,, 



^^ ^^ On, Qn . Qn-l-i, . . . Q» . Q«-l . . . Qi-&C. 



will be the Sturmian functions themselves, divided out by the 

 negative of the last or constant residue which arises in the appli- 

 cation of the process of continued division, according to Sturm's 

 rule ; and as we have shown that the series of the denominators 

 to the convergents of any continued fraction, and the series of 

 the denominators to the convergents of the same fraction reversed, 

 are signaletically equivalent, we have this surprisingly new, inter- 

 esting, and suggestive mode of stating Sturm^s theorem, viz. 

 the denominators to the convergents of the continued fraction 



fx 

 which represents '—- constitute a Rhizoristic series for f{cc), i. e, 



. . . J'^ . 



a signaletic series which serves to determine the number of roots 



of fx comprised within any prescribed limits. Moreover, in 



t^pplying this theorem it is by no means necessary that, in the 



fx 

 continued fraction which represents •^-tt-, all or any of the quo- 



tients should be taken linear functions of x. A very little con- 

 sideration of the principles upon which the demonstration of 

 Sturm's theorem is founded will serve to show that the conver- 

 gent denominators to any continued fraction whatever which 



fx 

 represents*'-^, whether the quotients be linear or non-linear, 

 jx 



