325 



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. cxlvi. (1856). 



VII. " Tables of the Sturmian Functions for Equations of the 

 Second, Third, Fourth and Fifth Degrees." By ARTHUR 

 CAYLEY, Esq., F.R.S. Received December 18, 1856. 



(Abstract.) 



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 hi 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), I 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. 



2 c 2 



