Mr. A. Cayley on the Sturmian Functions. 219 
roots, which vanishes when any two roots are put equal to each other, 
and that consequently such function expressed in terms of the coeffi- 
cients and equated to zero, gives the condition for the existence of a 
pair of equal roots. And it was remarked long ago by Professor 
Sylvester, in some of his earlier papers in the ‘ Philosophical Maga- 
zine,’ that the like method could be applied to finding the conditions 
for the existence of other systems of equalities among the roots, viz. 
that it was possible to form symmetric functions, each of them a sum 
of terms containing the product of a certain number of the differences 
of the roots, and such that the entire function might vanish for the 
particular system of equalities in question; and that such functions 
expressed in terms of the eoefficients and equated to zero would give 
the required conditions. The object of the present memoir is to 
extend this theory, and render it exhaustive by showing how to form 
a series of types of all the different functions which vanish for one 
or more systems of equalities among the roots; and in particular 
to obtain by the method distinctive conditions for all the different 
systems of equalities between the roots of a quartic or a quintic equa- 
tion, viz. for each system conditions which are satisfied for the particular 
system, and are not satisfied for any other systems, except, of course, 
the more special systems included in the particular system. The 
question of finding the conditions for any particular system of equali- 
ties is essentially an indeterminate one, for given any set of functions 
which vanish, a function syzygetically connected with these will also 
vanish; the discussion of the nature of the syzygetic relations 
between the different functions which vanish for any particular system 
of equalities, and of the order of the system composed of the several 
conditions for the particular system of equalities, does not enter into 
the plan of the present memoir. I have referred here to the indeter- 
minateness of the question for the sake of the remark that I have 
availed myself thereof, to express by means of invariants or covariants 
the different systems of conditions obtained in the sequel of the 
memoir; the expressions of the different invariants and covariants 
referred to are given in my ‘‘Second Memoir upon Quantics,”’ Phil. 
Trans. vol. exlyi. (1856). 
“Tables of the Sturmian Functions for Equations of the Second, 
Third, Fourth and Fifth Degrees.” By Arthur Cayley, Esq., 
F.R.S. 
The general expressions for the Sturmian functions in the form of 
determinants, are at once deducible from the researches of Professor 
Sylvester in his early papers on the subject in the ‘ Philosophical 
Magazine,’ and in giving these expressions in the memoir ‘ Nouvelles 
Recherches sur les Fonctions de M, Sturm,” Liouville, t. xiii. p. 269 
(1848), [ was wrong in claiming for them any novelty. The 
expressions in the last-mentioned memoir admit of a modification by 
which their form is rendered somewhat more elegant ; I propose, on 
the present occasion, merely to give this modified form of the general 
expression, and to give the developed expressions of the functions 
in question for equations of the degrees, two, three, four and five. 
