of Sturm's Theorem. 15 



is evidently the number of positive terms in the series 



111 1 



—> — i — i ~ i 



1 



/*1 /*2 



A*l' 11 



Pi 



These terms may be found with the utmost facility in succes- 

 sion from one another ; for if M t be one of them, the next will 

 be (/x-i+i — Mi) -1 . Thus, then, the necessity for the more operose 

 set of multiplications is done away with, and the actual labour 

 of computation reduced much more than 50 per cent, below that 

 required by the method indicated in the preceding article on the 

 subject. I need hardly add, that the old method of Sturm 

 would admit of a similar abbreviation ; but in using it we should 

 be subjected to the great practical disadvantage of having to 

 begin with the more heavy and complicated quotients fjb n> fjL n -i> 

 &c. instead of fi v /&%, &c, which would very greatly enhance the 

 labour of computation. I will conclude by a remark of some 

 interest under an algebraical point of view. 



It has been stated that the denominators of tjie successive 

 convergents to 



q n 1_ 



q n -\ 1 



<7i 



are equivalent (to a constant factor pres) with the Sturmian 

 functions, and the reader may be curious to know something of 

 the nature of the signaletically equivalent series formed by the 

 denominators of the convergents to the direct fraction 



These denominators are (abstracting from a constant factor not 

 affecting the signs) the Sturmian residues resulting from per- 

 forming the process of common measure between fas and fx ; 

 f x x being related in a remarkable manner in point of form to fx. 



