536 MR. SYLVESTER ON AN UNLIMITED ARBITRARY ALGEBRAICAL 
of terms from each of another set, by writing the former under the latter, and calling 
the (M roots of 0(x), and K the m roots of F{x), and F being sup- 
posed respectively of [m and m dimensions in x), and forming the disjunctive equations 
4, ^3 2, 3, y. 
3, m, 
we have the following equation, 
Q.^J= K„, , X sj {(pne^.^ne ,. . • -pne.Y X UK-fK - • • -/^ J" 
X 
ne, •• 
X 
ht .. 
‘2 
1 
ht,„^i 
htu+‘>" 
..A. 
^9v + 2* 
r ^si 
% •• 
v[ 
-h. 
h,i . 
....hf 
_^0V+1 
■ ■ 
+ 
1 
-Atn_ 
and moreover the different values ot aepenuuig upuu ... 
breaking up .. into two parts u and , are all (to a numerical factor fves) equal to one 
another. Thus then the theorem pointed at in art. (p.) is discovered, and the way 
laid open (by an unexpected channel) for a complete discussion of the theory ot he 
ulareases which may occur in the expansion of any rational algebraical fraction 
under the form of a continued fraction. j 
Art. Cl.). In the above expression, if we suppose ^,= 1, we have u-\ and v-O, or 
M=0 and ^=1, and remembering that 
h 
= (ph and 
K 
hi . . . ■ /l,; 
=FA 
nsi 
= ^'h, 
=:F'A, and 
Q,„ becomes by virtue of the general formula representable under either of the equi- 
valent forms . 
K.. , and K,. being either equal, or differing only in the sign agreeably to the formulse 
^ a”, n.). It may he worth while to notice, that, although (of course) these foriuulie 
and the general formulae of (art. 3.), when supposed converted into functions of -t- “ 
of the coefficients of F and of O by the reduction, integration and siimiiiation o i 
svmmetrical functions of the roots which enter into them remain universally valid, 
a'nd subject to no cases of exception, yet antecedently to these processes bouig pei- 
forined the formnlm as they stand may become illusory when any ^‘‘‘f'ons of eq™ 
exist between the roots of 4> inter se, or between the roots of F, liter se^ 1 
case before us, if <t- have equal roots the formulae eoini.iencing w h K,. is lUusoiy, 
and if F have equal roots the other of the two formulae becomes illusory, 
t Let us take the second of these and suppose that F(^) has 
hi roots c„ ^2 roots c.^, — /enroots Cp, 
