722 
ME. A. CAYLEY OX THE SYM:METEIC ELTYCTIOXS 
4bgh— 2abf 
4chf —‘lhQ,g = z\x,f^z^-\-zyc^y\z^, 
4afg — 2 cab=a^ 3 /iZ|^ 2 +^ 3 / 22 i' 3 ^iT 
4cgh — 2acf = yxz\o^^z^-\-y^z^^^z^, 
4ahf — 2bag=: z^a^{y\x^-\- z^^\x^, 
4bfg — 2cbh= x,y\zly^+x^ylz\y„ 
8f 2g _ 4chf — 2bcg = z\x^yl + z^^^y\, 
8g^h — 4afg — 2cah= 
8h^f — 4bgh — 2abf = y\z^0(^^-\-ylz^3(^^, 
8fg2 _ 4chg — 2acf = z\xiy.,-{- z^^\y^, 
8gh^— 4afh — 2bag= x\y\z^-\-x^^y\z^, 
8hP — 4bgf — 2cbh= y\z\x.^-\-y\z\x^. 
8 P — 6 bcf = y\z\-\-y%z\^ 
8 g® — 6 cag = z\x?^ + zlafi, 
8h® — QdibYi—xlyl -{-xly]. 
Secondly, consider the system of equations 
(«, h, c,f, g, A, ^,y, Jc, lX^, y, zf^O, 
(os, y'Xj^^ V") 
where the cubic function Avritten at full length is 
€13? -\-by'^-\-cz^-{- ^fy^z + 3 gz^x +3/? x^y + Myz^^ + ojzo? + 2>kxy- + 6 Jxy 
Joining to the system the linear equation 
(?, y. ^)=o, 
the linear equations give 
x\y\ z=^^ — yjj : y| — : ari — 18 |, 
and the resultant is 
(a, b, c,f, g, h, ij, k, lX^^—7n, y|— ati—(5^y=0, 
which may be represented by 
(a, b, c, f, g, h, i, j, k, 1 J|, 
where the coefficients a, b, &c. are given by means of the Table : — 
