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150] 
465 
150. 
A MEMOIR ON THE CONDITIONS FOR THE EXISTENCE OF 
GIVEN SYSTEMS OF EQUALITIES AMONG THE ROOTS OF 
AN EQUATION. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlvii. for 
the year 1857, pp. 727—731. Received December 18, 1856,—Read 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 Sylvester, 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 
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 coefficients 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 equation, viz. 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 
c. II. 59