EEY. E. HAELET ON THE METHOD OF STMMETEIC PEODHCTS. 
329 
then, combining the conditions of symmetry, and rejecting those values of k (k^ or ks 
indifferently) which render X symmetric, we find 
k^-i-k-j-l=0, 
that is, k is an unreal cube root of unity. Eepresent this root by a, and put 
— Xi-\- Ci 
fis^ ) — Xi~j~oi X2~i~c( X 2 , 
•f{K')={%xy—^'%x,X2=^{b'^— ^ac). 
If, in evolving the roots by this method, we replace /(a) by/, and by there will 
/ — Xi~\~Ci X 2 ~\~cc . 23 , 
.21 
j=cCi-\-a X2 + a X3, 
then 
and 
result 
so 
that 
^2~\~ X2 , 
/+7=h^+S“-.). 
and therefore 
/»+^=i(l, -3}, 3*«c, -SVd^b, 1)» 
~ (-2b^+9abc-27a\I); 
whence, solving as for a quadratic in /^, restoring the value of w, and extracting the 
cube root, we have 
f = — 2 7 a^d •\-9abc—2b^-\- 3«a/ 3(27 — 1 %abcd + -\-Wd— J | , 
and the roots of the complete cubic are included in the formula 
where m=l, 2, or 3. 
In my original memoir I followed Mr. Cockle, and in the application of the theory 
to the solution of the cubic and the biquadratic, I employed a subsidiary equation of 
the same degree. This equation was obtained by eliminating x between the given one 
and 
y—^{x)=9, 
2 z 2 
