OF PARTIAL FRACTIONS INTO A CONTINUED FRACTION. .541 
it is therefore now known as a rational and integral (uactSon of*; h,h,...h,-, c,c,...c 
The allotrions factor itself is made up of the product of squares of qnantitierallo’f 
the same form as the leading coefficient in D,*, which, from what has been shown 
above, is seen to be equal to 
Hence each terra in the continued fraction 
1 1 
(Aia7 + Bi)- (A2 x + B2)^'”(A„^+B„)’ 
which is to be made equal to 
^1 I ^^2 I Cn 
{x-h^) “1“ [x-h^) 
IS completely assigned in terms of x and the given quantities c and h. 
Art. (D.). The number of effective intercalations between the roots of F^r is easily 
seen to be equal to the excess of the number of positive real numerators over the 
number of negative real numerators in the partial fractions of which is the sum 
and hence we see h priori, as an obvious consequence of a simple extension of the 
reasoning in art. (47.), that the inertia of the quadratic function 
sj -f- +....+ 
Where Cg=^^ wdl represent the value of the index in question. So too we may 
see that the formula given for the residues to fxj'x in art. (46.) continue to apply 
to the residues Fj-, ^x. That is to say, these residues when divided out by Fd? will be 
respectively represented by the successive principal coaxal determinants to the matrix 
SflSi S 2 
S1S2 S3 ...s,„ 
S2S3S3 ...s. 
S^^J . . .S2,„_5 
' X — hy 
where in general 
and using the same matrix as above written with S' substituted for S, where in general 
S,=c,(*— A);+c,(*— 
the successive principal coaxal determinants of the new matrix represent the succes- 
sive denominators to the convergents of the continued fraction which expresses 
The expression for the numerators to the convergents may also, there is no doJbt, 
e obtained by some simple modification (dependent on introducing the quantities 
CiC2...c„) of the formula in art. (41.), p. 465. 
I annex, more with the hope of suggesting than (in all instances) of conveying a 
