OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
487 
and in general 
(-) 
^p + q- 
> = 2 , (-)-> - 
I 
l)\ q 
9. Moreover, the fundamental symmetric functions are expressed in the terms of the 
forms Spq by the formula 
a„ = 2 ^ A+fyliy j(ii±&pir 1 '■ 
^ ' i^i •' ?Zi! J L Ih •' 72! ■ • • 
(-) 
StT — 1 TTi 77.I 
IT 1 IT a . 
as will be evident by simply applying the multinomial theorem to one of the above 
written general identities of Art. 8. 
10. The single-bipart functions having been actually expressed in terms of the 
fundamental symmetric functions, it remains to show that all other rational algebraic 
symmetric functions are also so expressible. Schlafli (Joe. cit.) has established this 
by induction, and it is not necessary to further discuss the theorem here. In Art. 43 
of the present memoir, will be found the actual expression of a given symmetric 
function by means of single-bipart forms, a formula which, combined with one given 
above. Art. 8, serves to establish the theorem conclusively. 
11. Write 
The Symmetric Function hprp 
(1 + ayr + ^yj) (1 + a,x + ^.yj) . . . = 1 + + 
1 
+ ap^xPy'i + . . . 
1 — — JiQ^y + . . . + . 
as the definition of the function 
Writing — x. — y for x, y, we have 
1 -f h^yx + h^y + . . . + hp^xJPy^ 
(1 - a^x - /3^y) (1 - a^x - . . 
and expanding the right hand in ascending powers of x and y 
(Pi + < 7 i) ! (Pa + 72 )! 
hpq - S 
Pi \qy. P 2 ! !72 
I ru ! 
• • {lh 9 .i • • •)> 
the summation being for all partitions of the biweight. Changing the signs of x and 
y in the relation first written down, we obtain 
1 + h^^x + h^^y + . . . + hp^xPyi + • • . = 
-r . . . -f- (—+ ? ap^Pyi + . . . 
