fx 
OF ^ UNDER THE FORM OF AN IMPROPER CONTINUED FRACTION. 471 
and Pj-, which is the ^th simplified residue, vanishes when the n roots in any manner 
become reduced to only i distinct groups. 
I proceed to show that if we make 
A,a?+Bj=Ui=A? Ai)+A? 2 (^ 7 — ^ 2 ) + ... +A| — h„), 
where in general 
A,,, represents 
then will 
T,=U,. 
It will be observed that A,- ^ is identical with what the simplified denominator of the 
(^— l)th convergent becomes when we write h, in place of x, and consequently, when 
arranged according to the powers of Ji„ will be of the form 
where c„ Cg, ... are functions of the coefficients, but containing no more of them 
than enters into Qi_i, i.e. containing only 2i—2 of them. 
Now A; is made up of terms, each consisting of some binary product of 
Cl, C2, . . .5 Cl 
combined with some term of the series 
... 
and any one of this latter set of terms expressed as a function of the coefficients of fx 
contains at most 2i — 2 of them. 
Hence only 2i—2 of the coefficients enter into A„ and in like manner only 2i—l 
of them into B^. 
The number of letters, therefore, in A; and in B^ is the same as in and in M;, 
viz. 2i—2 and 2i—\ respectively. 
Now let the roots consist of only i distinct groups of equal roots, so that 
becomes 
I shall show that in whatever way the equal roots are supposed to be grouped upon 
this supposition, there will result the equation 
= 2 {% • • •%» %2 • • • %j-l) } 
Hj= Af .;?i+A 2 .%+ ... +A^.;j„, 
^{{ne—nefne— %) 1 ) } , 
and 71^ meaning x—h^. 
MDCCCLIII. 3 Q 
where 
and 
Ag meaning 
