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VII. Memoir on Symmetric Functions of the Roots of Systems of Equations. 
By P. A. MacMahon, Major, Royal Artillery. 
Communicated hy Professor Greenhill, F.R.S. 
Received January 30,—Read February 6, 1800. 
§ 1. Preliminary Ideas. 
1. The theory of the symmetrical functions of a single system of quantities has 
been investigated in a large number of memoirs, but so far, only a few attempts have 
been made to develop an analogous theory with regard to several s 3 ^stems of 
quantities. The chief authors are Schlafli* and Cayley,! both of whom have, 
however, restricted themselves to the outlines of the commencement of such a theory. 
In the theory of the single system it is found convenient to regard the quantities as 
the roots of an equation, since the coefficients of such an equation are themselves 
those particular symmetric functions of the quantities which have been variously 
termed fundamental, elementary, and unitary; they are fundamental because all 
other rational integral functions are expressible by their products of the same or 
lower degree ; elementary because they are those which, first of all, naturally arise ; 
unitary because their partitions are composed wholly of units. The left hand side of 
the equation referred to is a product of binomial linear functions of a single variable 
X, so that, «i, a^, . . . being the cjuantities which compose the system, the funda¬ 
mental relation may be written 
(1 + (1 + %*) •••(!+ = 1 + ayx + a^yF + . . . + j 
= 1 + (1) X + (1^) + . . . + (1") x" , 
in the ordinary partition notation. 
In a general disoussion it is convenient and advantageous to suppose the number of 
quantities infinite, so that the relation becomes 
(I -f ayx) (1 + ccyx) . . . = (l fi- ayjc -b a^x^ + . . .) = 1 + (l) a: -j- (1") 
* “ Ueber die Resultante eines Systemes mebrerer algebraiscben Gleicbungen.” ‘ Vienna Academy 
Benhscliriften,’ vol. 4, 1852. 
t “ On tbe Symmetidc Functions of the Roots of certain Systems of Two Equations.” ‘ Phil. 
Trans.,’ vol. 147 (1857). 
MDCCCXC. —A. 3 Q 
19.9.90 
