480 MR. SYLVESTER ON FORMULAE CONNECTED WITH STURM S THEOREM. 
roots of/r K tbe theorem already cited in a preceding article, concerning the 
denominators of the convergents, may be expressed as follows ; 
(©■■ -■ (£■)■ ' 
(T./i,)”; (T. 
(T, (T, /!,)"; (T, S,)’ 
where it will be observed that the first line of terms consists exclusively of units. 
since f'x=(px by hypothesis. ^ _ 
Correlatively I have ascertained that preserving the same assumption that ^x-f x. 
so that consequently ^ means the following theorem obtains, viz. that if 
a%y. 
1 
Ui) ’ 
{t 2 {h)y ; 
...{u{kn^i)y 
(^- 2 (^ 2 )) 5 
...{tn-2{kn-^)y^ 
It may consequently be conjectured, when <p and / are independent functions of 
X and respectively of the degree «-l and «, and | is expanded under the form of a 
continued fraction, of which, as before, are the successive convergents. 
'that we shall have analogous determinants to the twin forms above given, each 
separately vanishing, these more general determinants differing only from tlieir 
model forms in respect of the uppermost line of terms m the one of them, being 
each multiplied by certain functions of A., h.,...h. respectively (all of which become 
units when <px=fx), and in the other of them by certain functions of A,,...A„. 
The exact form, however, of such functions, and even the possibility of such form 
being found capable of making the determinants vanish, remains open for further 
inquiry. 
Section IV. 
On some further Formulae connected with M. Sturm’s theorem, and on the Theory of 
Intercalations whereof that theorem may he treated as a corollary. 
As preparatory to some remarks about to be made on the formulae connected with 
M. Sturm’s theorem, it is necessary to premise two theorems concerning quadiatic 
functions of great importance, one which, notwithstanding its extreme simplicity, is 
as far as I know very little (if at all) known, and the other was given m part many 
years ago by M. Cauchv, but is also not generally known. The former of these 
