147] 
417 
147. 
A MEMOIR ON THE SYMMETRIC FUNCTIONS OF THE ROOTS 
OF AN EQUATION. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlvii. for 
the year 1857, pp. 489—499. Received December 18, 1856,—Read January 8, 1857.] 
There are contained in a work, which is not, I think, so generally known as it 
deserves to be, the “ Algebra ” of Meyer Hirsch [the work referred to is entitled 
Sammlung von Beispielen Formeln und Aufgaben aus der Buchstabenrechnung und 
Algebra, 8vo. Berlin, 1804 (8 ed. 1858), English translation by Ross, 8vo. London, 
1827] 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 (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 
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 (6), 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, odf = (1 — ax) (1 — ¡3x) (1 — yœ) ..., 
so that 
— b = 2a, + c = 2a/3, — d = Xafiy, &c., 
and if in general 
C. II. 
(pqr...) = Xa p /3 q y r ..., 
53