474 MR. SYLVESTER ON THE QUOTIENTS RESULTING FROM THE EXPANSION 
of a rational function of x by another one degree lower, introduces into the integral 
part of the quotient the square of the leading coefficient of the divisor, subject to the 
exception that when the divisor is of the degree zero, the simple power enters m lieu 
of the square. The general formula gives for the reduced nth. quotient the ex- 
pression 
2 { — K — h . • • ^ hJjhh-‘-hn))\x~hj), 
which equals 
Rejecting the first factor, we have 
lXjh,K...K){x-hj, 
which is equal to the penultimate residue, which residue is (as it evidently ought to be] 
identical with the simplified last quotient. 
• c 
Art. (i.) We have thus succeeded in giving a perfect representation of i. e. ot 
_J_. . . ^ 
x—hj'x — hj' ' x—h^i 
under the form of a continued fraction of the form 
I 1 1 
rwj [x — ed — m^{x—e^ — m^ix — 
where ; e, e^...e„ are all determinate and known functions of 7q h....h„. 
We may by means of this identity, differentiating any number of times with respect 
to X both sides of the equation, obtain analogous expressions for the series 
1 . 1 , . I_ 
{x—h^Y^{x — h^Y^ ^{x—k„Y‘ 
But to do this we must be in possession of a rule for the differentiation of continued 
fractions whose quotients are linear functions of the variable. I subjoin here the 
first step only toward such investigation. 
Let the denominator of 
J 1_ 
qi- q^-'” ' q» 
where ai’e any n arbitrary quantities, be denoted by [^i, q., that 
the entire fraction will be equal to 
[(Za ■ •?»] 
Iqi 
any such quantity as may be termed a Cumulant, ofwhich qi, qi+.-.-q,, may 
be severally termed the elements or Components, and the complete arrangement of the 
elements may be termed the Type. The cumulant corresponding to any Type remains 
unaffected by the order of the elements in the Type being reversed, as is evident from 
any cumulant being in fact representable under the form of a symmetrical determmant. 
