2 REPORT — 1853. 



f'x 

 If we suppose - — , by means of the common measure process, to be expanded 



under the form of an improper continued fraction, the successive quotients will be 

 the values of Qj Qj . . . . Qn above found. 



The successive convergents of this fraction will be 



JL Qa Q0Q3-1 . A 



Qi' Q,Q,-1 ' QAQ3-Q1-Q3' ' fa' 



The numerators and denominators of these convergents will consequently also be 

 functions of the factors exclusively. They are the quantities the sum of the pro- 

 ducts of which multiplied respectively by fx and f'x produce (to constant factors 

 ^res) the residues. The denominators are expressible very simply in terms of the fac- 

 tors and the differences of the roots ; and their values under such forms were pub- 

 lished by me about the same time as the values of the residues in the Philoso- 

 phical Magazine ; the expression for the numerators is much more complicated, but 

 is given in my paper, " The Syzygetic relations," &c., in the Philosophical Trans- 

 actions. 



By comparing the expression for any quotient with the expressions for the two 

 residues from which it may be derived, we obtain the following remarkable identity : — 



Z._jXZ,., i.e. ^^(hji,....hi_,)x:zi:(h,L....ki) 



■=iP' + iPl + iPl+ +,P„- 



When the roots are all real, we have thus the product of one sum of squares by the 



product of another sum of squares (the number in each sum depending upon the 



arbitrary quantity t), brought under the form of a sum of a constant number (n) of 



squares, which in itself is an interesting theorem. 



The expression above given for Qj leads to a remarkable relation between the 



f'x 

 quotients and convergents to •'- — 



Let it be supposed, as before, that 



fx_ I 1 _J. 1 



fx Q,«— Q«a?— Qs*— Q„(*)' 



and let the successive convergents to this continued fraction be 



N.Cx) ^,{x) NaCa;) N„(a;) 



D,(a;)' D,{x)' D,{x) D»' 



where the numerators and denominators are not supposed to undergo any reduc* 

 tions, but are retained in their crude forms as deduced from the law 



N,=Q, N,_,-N,_„ 



D,=Q,.D,_,-D,.,. 



Ni (x) being 1, and D, (x) being Q, (x), then it may be deduced from the pub- 

 lished results above adverted to that 



D.W= ^r 5"' ^ {K{hM-hed {x-he,) (x-he.,) ...(x-he,)}. 



Hence 



ry2 "72 '72 



= ^■•-'- '•-=' ^'-'^+' D,_, (K) ; 



