452 MR. SYLVESTER ON THE RESIDUES AND SYZYGETIC MULTIPLIERS TO <px, fx 
(IS functions of the roots of the given functions ; but if in art. (16.), instead of writing 
we had made 
1 
L 
2 * * * 
K • 
..K “ 
X 
^'*9u + 2 • 
.,h„ 
J 
‘ ■ %2' * 
■ 
hq. •• 
X “ 
HV 
_ ^qv+\ 
+ ■ 
X 
X 
_ ^^1-+ 1 ^#J/ + 2 ■ ■ ■ ^fTTi _ 
where P represents any function symmetrical in respect of and also in 
respect of (the interchanges, that is to say, between one h and another h, 
or between one and another ??, leaving P unaltered), it might be shown that the 
value of resulting from the introduction of this more general value of R would 
(as for the particular value assumed) always be expressible as an integral function of 
the roots, and consequently, if P be taken of the same dimensions in the roots as the 
numerator of R previously assumed, i. e. w, \ ^ would continue to be (unless indeed 
it vanish) identical (to some numerical factor jore^) with the corresponding simplified 
residue. If, on the other hand, P be taken of less than vv dimensions in Ji and k, 
we know a priori that „ must vanish, as otherwise we should have a conjunctive of 
a weight less than the minimum weight. When P is of the proper amount of weight 
vv, it is I think probable that another condition as to the distribution of the weight 
will be found to be necessary in order that „ may not vanish, viz. that the highest 
power of any single (A) in P shall not exceed v, nor the highest power of any single r, 
exceed v. But as I have not had leisure to enter upon the inquiry, the verifiC(ation or 
disproval of this supposed law, and more generally the evolution of the allotrious 
numerical factor introduced into by assigning any particular form to (P) satisfying 
the necessary conditions of amount and distribution ot weight, must be reserved, 
amongst other points connected with the theory of the remarkable forms (19.) art. (15.), 
as a subject for future investigation. 
Art. (31.). A property of continued fractions, which, if known, I have not met 
with in any treatise on the subject (but which has been already cursorily alluded to 
in these pages), gives rise to a remarkable property of reciprocity connecting r and t 
severally with ^ in the syzygetic equation rf—t(p^'^—0. 
Let the successive convergents to the ordinary continued fraetion 
1 1 1 1 I 
<li+ 9^+ 9'3+ 2t-i+ Qi 
be called 
Z] Zg Zj—J Z,' 
m,’ uii 
respectively, it is well known that 
