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XLIV. On the explicit Values of Sturm's Quotients. 

 By J. J. Sylvester, F.R.S.* 



BY Sturm's quotients is of course meant to be understood 

 the quotients which result from applying the process for 

 the discovery of the greatest common measure between fx (an 

 algebraical function of the nth. degree in x, and whose first co- 

 efficient is unity) and fx its first derivative, as in Sturm's 



f x 

 theorem ; or which is the same thing in effect, supposing J -j- to 



Jx 



be represented by 



_1 1 I I 



Qi- 02- Qb- " ' 0.' 



(where Q, v Q 2 , . . . Q ra are all linear functions of a?), the quotients 

 in question are Q 2 , Q 2 , . . . Q n . Before proceeding to discuss 

 these quotients, it will be well to state the form under which the 

 other quantities which appear in the course of the application of 

 the Sturmian process admit of being represented. First, then, 

 it will be remembered that the residues with the signs changed 

 are all of the form 



E J =MpS{f(A 1 , h 2 ,...h t )(x-h i+l ) (x-h i+2 ) . . . (a? -£„)}, 

 where %(h v h^ . . . h t ) indicates the squared differences between 

 every two of the quantities h } , k 2 , . . . h ( and h v h^ . . . h n are sup- 

 posed to be the n roots of fx ; and where using f. to denote 

 %£(h v h^, . . . h t ), with the convention that f =l, K\ — n > an d un- 

 derstanding by (i), {i+(— y}^ 



yi 9*2 y 



M = z$-l ' *~ 4 ' ' * b ^ +1 

 i yi yi ?2 



fei-l * 5£-3 ' * * »(t) 



Here it will be observed that the only quantities appearing 

 are the factors and the differences of the roots of fx ; and since 

 these latter are the same as the differences between the corre- 

 sponding factors (for (x— h) — (x— h') = # — h), the entire quan- 

 tity which expresses any residue R^ may be considered as a func- 

 tion of the factors of fx exclusively. 



Again, if we solve the syzygetic equation 



I have published many years ago in this Magazine the value of D b 

 and subsequently in a paper read before the Royal Society on the 

 16th of June last thevalueof N p both which values are also functions 



* Comnmpicated by the Author. 



