q/ Sturm's Theorem.-' '^: ^ " ''' 447 



two series being signaletically equivalent when the number of 

 continuations of signs and of variations of signs between the 

 several terms and those that are immediately contiguous to them 

 is the same for the two series ; a condition which evidently may 

 be satisfied without the order of such changes and continuations 

 being identical. I am now able to enunciate the following 

 remarkable theorem of signaletic equivalence between two distinct 

 series of terms, each generated from the same improper continued 

 fraction. But first I must beg to introduce yet another new term 

 in addition to those already employed, viz. reverse conver gents, to 

 denote the convergents generated from a given continued frac- 

 tion by reading the quotients in a reverse order, or if we like so 

 to say, the convergents corresponding to the given continued 

 fraction reversed. 

 The two forms 



,^^-^_-^ and -^^ 



g'2 1 fi'n-.l 1 



^3 5'«-2^ 



• 1 



1 



are obviously reciprocal ; and if the two last convergents of either 

 one of them be respectively 



"~^ will serve to generate the other. For the clearer and more 



simple enunciation of the theorem about to be given, it wil] be 



better to take as our first convergent j, so that 1 will be treated 



as the denominator of the first convergent in every case ; and 

 calling Dq such denominator, we shall always understand that 

 Do=l. Let now D^, Dj, Dg, ... D^ be the [n+l) denominators 

 of any improper continued fraction of n quotients, and (Jq/ (Ti> 

 da^ • • • • Qn the corresponding denominator series for the same 

 fraction reversed ; then, I say, that these two series are signaleti- 

 cally equivalent, 



I do not here propose to demonstrate this proposition, to 

 which I w^as led unconsciously by researches connected with the 

 theory of elimination, which afford a complete and general but 

 somewhat indirect and circuitous proof. Doubtless some simple 

 and direct proof cannot fail ere long to be discovered*. For 



* See Postscript. • 



