724 
ME. A. CAYLEY ON THE SY3DIETEIC FUNCTIONS 
distinct terms obtained by permuting the different sets of roots, so that the equations for 
the fundamental symmetric functions are — 
a = 
b= VMz. 
C — Z2 
3 f = 83/1^2^3, 
3 g= 8^1 22^3, 
3 h= 8 ^1^23/3, 
3 i =8yi 2:22:3, 
3 j =8 2:1 
3 k =8 ^,3/2^3, 
61 — 8 ^1^2 ^ 3 , 
then the complete system of expressions for the symmetric functions of the second order 
is as follows, viz. — 
p ^ i /j.»2 
Cv tv j tv 2 *^ 3 5 
P 2 __ «2 ^2 
C -^1 -^2 
bc= ViZ.y^z^ij^z^, 
ca= ZiX^z^x^ZgXs, 
ab = ^ iy <iy •it 
3 af x^yiX^y-iZiX^^ 
3 bg= 8 3/12,3/22:2^3^3, 
3 ch = 8 A’l z^x^yz 2:3, 
3 bf = 83/^3/^3/323, 
3 cg =8 2:^212:3^3, 
3 ah = 8 x^iXlXsys, 
3 cf =83/1 z.y^z^zl, 
3 ag = 8 2:, a’, z^x^xl, 
3 bh ^^8 X ^y 2y -2 yat 
3 ai 8 X ^y \Z2^'2^^3*^ it 
3 bj =S y.z.x^y^^'iyit 
3 ck =8 ZiX^y^z-iyiZ^. 
