of Sturm's Quotients. 295 



in itself a truly remarkable equation, which it will be seen is of 

 2(i — l) 2 dimensions in respect of the roots*. 

 When i=l, 



^=2(07-^); 

 and when i=2, 



T,-*[(*<* 1 -*,)) , }(»-iA 



i. e. »2{((n-l)^-(Vt-*8+ • • • +*j)V~*,)}- 



When i=n, T n becomes 



as it evidently ought to do. Substituting for T<-i, T. and A., 

 their values, we have as the complete general expression of the 

 ith Sturmian quotient the following expression, in which, agree- 

 able to a notation which I have previously used and explained, 



[h 9i h 0m . . . *i. ] means (*i-^)(*i--^ * * * ( A i- V> 



'2 *i-l 



viz. 



*2 f4 £4 .£4 4- 



O = *~* *'~ 3 * fri-5 * * * WO ' x 



^* 5*2 ' £4 f4 £4 



±i bj_2 * bj-3 ' ' ' »(0 + l 



2{{S(?(A«, * . . . ^_ i} [^ ^ . . . V J) }>-*,)}• 



It ought not to be passed over in silence, that if we write 

 _L 1 1 1 _N,M 



Qi- Q 2 - Qs- '"Qi D«W i 



and if we suppose N 4 (a?) and Dj(a?) to be expressed integrally, 

 and to be algebraically prime to one another, then 



D i -,w=s{r^^...y i )[; i , v ,,^ i ]}. 



* Thusifw=4andi=2 &-i=4 f 2 =S(A,— Aj) 2 , 

 and we have 



4{(A 1 -A 2 )24-(A J -A 3 ) 2 +(A 1 -A 4 ) 2 4-(A 2 -A 3 )24-(A 2 -A 4 )2 + (^-^} 



=(3A 1 -A 2 -A 3 -A 4 ) 2 -|-(3A 2 -^-^-A 4 )2 + (3A 3 -A 1 -A 2 -A 4 ) 2 



and so in general f 4-1 . £\, which is the product of two sums of variable 

 numbers of squares, is expressible rationally as the sum of a constant num- 

 ber (w) of squares for all values of i. 



t (t) denotes *{(-!)*+!}. 



