294 Mr. J. J. Sylvester on the explicit Values 



N 

 of the factors of fx exclusively. _J', it is easily seen, represents 



the successive convergents to the continued fraction by which 



f*x 



J -z- is supposed to be expressed, and B, (to a constant factor 



pres) is the denominator of the reverse convergents of the same 

 continued fraction. To the completion of this part of the theory 

 it evidently therefore becomes necessary to express the quotients 

 Qv Qa> Qs> • • • Qn-i Q n (of which the first (n — 1) are those 

 which appear in Sturm's process, and the last is simply the 

 penultimate Sturmian residue divided by the ultimate residue) 

 under a similar form, i. e. as functions exclusively of the factors 

 of fx, or, which comes to the same thing, of the factors and the 

 differences of the roots. Guided by an instinctive sense of the 

 beautiful and fitting, in a happy moment I have succeeded in 

 grasping this much wished for representation, with which I pro- 

 pose now and for ever to take my farewell of this long and 

 deeply excogitated theorem. 

 If we write 



R,._ 1 = M t _ 1 {A i _ 1 a?»- l+, -B i _ 1 ^- l +&c.}, 

 and 



R.=M.{A i tf»-*-B. < z*- t - 1 + &c.}, 

 we have 



A^^Sr^V-.^-i) B,_ 1 =2(A, + A <+1 + ...+h n )i{h l h 2 ...h i -i) 

 A.=Sr(^ A 2 ... h.) B i =2(h i+1 + h i+2 +... + h n )Z(h l £..*<); 

 and the ith quotient is evidently 



M t _! A f _ 1 A t ^ + (A t - 1 B.-A i B t _ 1 ) 

 M< J* ' 



and this is the quantity (unpromising enough in aspect) to be 

 transformed in the manner prescribed. 



Mi_!, M £ , and A 4 are already given under that form, and I 

 find that, putting 



T.= A,., . A.# + (A f _! . B.-A.B^O, 

 Tj may be represented by the double sum 



This of course implies the truth of the identity 



2{2f(A e , A flj . . . ^(A,-*^*!-**) • • • (A.-V,)} 2 

 =Ai_, .A,asfc.,.fc, 



