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XXIII. A Memoir on the Symmetric Functions of the Roots of an Equation. 
By Aethue Cayley, Esq., F.R.S. 
Eeceived December 18, 1856, — Bead January 8, 1857. 
Theee are contained in a work, which is not, I think, so generally known as it deserves 
to be, the ‘ Algebra’ of Meyee Hiesch, some very useful tables of the symmetric functions 
up to the tenth degree of the roots of an equation of any order. It seems desirable to 
join to these a set of tables, gmng reciprocally the expressions of the powers and pro- 
ducts of the coefficients in terms of the symmetric functions of the roots. The present 
memoir contains the two sets of tables, viz. the new tables distinguished by the letter [a), 
and the tables of Meyee Hiesch distinguished by the letter (b ) ; the memoir contains 
also some remarks as to the mode of calculation of the new tables, and also as to a 
peculiar symmetry of the numbers in the tables of each set, a symmetry which, so far as 
I am aware, has not hitherto been observed, and the existence of which appears to con- 
stitute an important theorem in the subject. The theorem in question might, I think, 
be deduced from a very elegant formula of M. Boechaedt (referred to in the sequel), 
which gives the generating function of any symmetric function of the roots, and contains 
potentially a method for the calculation of the Tables [b), but which, from the example 
I have given, would not appear to be a very convenient one for actual calculation. 
Suppose in general 
(1, b, c..^l, = — — 
so that 
— Z»=2a, -f-C=2a|8, — d='2oi(5'y. See., 
and if in general 
(pqr . . . ., 
where as usual the summation extends only to the distinct terms, so that e. g. {p^) con- 
tains only half as many terms as {pq), and so in all similar cases, then we have 
-^=(1), +c=(H), -(Z=(r), &c.; 
and the two problems which arise are, first to express any combination ¥(f ... in terms 
of the symmetric functions {Jfm^..), and secondly, or conversely, to express any 
symmetric function {f m?...') in terms of the combinations b^c'^ .... 
It will conduce materially to brevity if H2’... be termed the partition belonging to 
the combination b^c'^ ...; and in like manner if I’m ? ... be termed the partition belonging 
to the symmetric function {fun? .. .), and if the sum of the component numbers of the 
partition is termed the weight. 
Consider now a line of combinations corresponding to a given weight, e. g. the 
weight 4, this will be 
MDCCCLVII. 3 s 
