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XXXIII. A MemoiT on the Conditions for the ^Existence of given Systems of Equalities 
among the Roots of an Equation. By Aethue Cayley, Esq., F.R.S. 
deceived December 18, 1856, — Head January 8, 1857. 
It is well known that there is a symmetric function of the roots of an equation, viz. the 
product of the squares of the differences of the roots, which vanishes when any two roots 
are put equal to each other, and that consequently such function expressed in terms of 
the coefficients and equated to zero, gives the condition for the existence of a pair of 
equal roots. And it was remarked long ago by Professor Sylvestee, in some of his 
earlier papers in the ‘ Philosophical Magazine,’ 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 con- 
taining the product of a certain number of the differences of the roots, and such that 
the enthe function might vanish for the particular system of equalities in question ; 
and that such functions expressed in terms of the coefficients and equated to zero would 
give the requhed 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 equation, \iz. for each system con- 
ditions 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 equalities is essen- 
tially 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 indeterminateness 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,’ Philosophical Transactions, vol. cxlvi. (1856). 
1. Suppose, to fix the ideas, that the equation is one of the fifth order, and call the roots 
a,(3,y,^,s. Write = 12.13=2®(«-|3)'(«-7)'", 12.34 = 2?)(«-^3y(y-^)“. 
&c., where 9 is an arbitrary function and I, m, &c. are positive integers. It is hardly neces- 
sary to remark that similar types, such as 12, 13, 45, &c., or as 12.13 and 23.25, &c., 
denote identically the same sums. Two types, such as 12.13 and 14.15.23.24.25. 34. 35.45, 
5 B 2 
