296 On the explicit Values of Sturm's Quotients. 



Hence Q, is contained as a factor in 



(D,., . *,)•(*-*,) + (D<_, . A s )*(*-A 8 ) . . . + (D ( _, . *.)«(*- A,). 



It may be observed also, that for all values of i between 1 and 

 n inclusively, 



D^ + D^ + DiAgH- . . . +1)^=0, 



and also that the determinant 



1 1 1 ... 1 



(DA)* (D A)» (D A) 2 . . . (DA) 2 



WW (DA) 2 (DA) 2 • • • (DA) 2 



(D.-i*,) f (D n -A)* (D„-^ 3 )2...(D n _A) 2 

 is always zero. To complete the theory, I subjoin the value of 



f f x 

 N|, the simplified numerator of the ith convergent to J -x-, 



J x 

 ■expressed as an improper continued fraction. 



Let the sum of the products of x—h,x—k, . . . x—l com- 

 bined i and % together be denoted by S { (A, k, . . . /), and the sum 

 of the 2th powers of the same by a^h, k, . . . /), then 



x s i(^e i+1 - A B ) + ^-3^61, \*»h) • S 2(^6> 1 . +1 — \) + &c - 

 ... t^ + l)^,^...^)}. 



The anomaly of the last term being of the form (1 +(r ) . S f _! 



(for of course <r =i), instead of being cr . S^, is not a little 



remarkable. 



Of the four sets of Sturmian quantities, viz. the residues, the 



quotients, and the denominators and numerators of the conver- 



f'x 

 gents to J - F - } it will have been seen that the first and third are 

 jx 



expressible in terms of the roots and factors by single summa- 

 tions of equal simplicity, the second and fourth by double sum- 

 mations, whereof that which corresponds to the numerators is 

 much the more complicated of the two. 



7 New Square, Lincoln's Inn, 

 September 3, 1853. 



