218 Royal Society :— 
these until we know more about them; but it may be as well to 
state, that if a mixture of phenic acid, acetic acid, and iodine be 
heated in a sealed tube, there is a product obtained producing 
salts of a similar colour to those obtained with bromoacetic acid ; 
and also if acetic acid be replaced by any of its homologues, 
similar acids are obtained, evidently showing the existence of a 
whole series of these acids. However, we expect soon to be able 
to give a full account of bromoacetic acid, and hope to obtain 
the bi- and tri-bromoacids, and also to give a complete account 
of the acids procured by treating phenic acid with bromoacetic 
acid. 
August 1857. 
XXVII. Proceedings of Learned Societies. 
ROYAL SOCIETY. 
[Continued from p. 153.] 
January 8, 1857.—William Robert Grove, Esq., V.P., in the Chair. 
HE following communications were read :— 
“Memoir on the Symmetric Functions of the Roots of an 
Equation.” By Arthur Cayley, Esq., F.R.S. 
There are contained in a work, which is not, I think, so generally 
known as it deserves to be, the ‘ Algebra’ of Meyer Hirsch, 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, giving reciprocally the expressions of the powers 
and products 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 
Meyer Hirsch distinguished by the letter (4) ; 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 constitute an 
important theorem in the subject. The theorem in question might, I 
think, be deduced from a very elegant formula of M. Borchardt 
(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 (4), but which, from the example I 
have given, would not appear to be a very convenient one for actual 
calculation. 
** Memoir on the Conditions for the Existence of given Systems of 
Equalities among the Roots of an Equation.”” By Arthur Cayley, 
Esq., F.R.S. 
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 
