330 
EEY. E. HAELEY ON THE METHOD OE SYMMETEIC PEODHCTS. 
being rational and so constructed as to make T(y) vanish. It is true that the eva- 
nescence of TT leads to an immediate solution, and that when y is known, A’ is also kno-wm. 
But this evanescence is not essential to the theory ; and we are conducted to more signi- 
ficant results by dispensing with it. 
4. For the quartic 
( a , b , G , d , djx , 1)^=0, 
assume (cyclically) 
Xj = -j- “b ^ 2^3 “b ^ 3 ^ 4 ? 
5^3 — “b 
and ‘r 3 (^) =Xi Xj X 3 . 
Then, as before, combining the conditions of symmetry and rejecting those values of k 
which would render X symmetric, we are led to the cubic 
1 = 0 , 
of which the roots are —1, 1 and —1. Let therefore 
then 
X j = — .T 2 +<^ 3 — ^45 
X 2 =a^i-ba^ 2 — ^ 3 — ^ 4 ? 
— ^ 3+^45 
r3(^) = XiX2X3= — i%x%x^X2-\- ^%x^x.^X3=.^ ( — + ^dbc— ^a^d). 
5. The following solution is due to Mr. Cockle, who communicated it to me in Sep- 
tember of last year. It may be considered as an extension of a solution of the complete 
cubic which I sent to him in January of the same year, and which, not essentially 
differing from the above, has a certain resemblance to that given by MuEriiT in the 
‘ Philosophical Transactions ’ for 1837. 
Let 
— Xi~~ x2~j~x^—x ^ , 
and 
then 
TT 
' — — — . /y» ■ . /y» __ /y* /m 
^ f 0/2 *^4 9 
y 1^2 
and as in cubics, 
/i +/ i— ^ ^ + 4aa^i), 
JlJ'2 “ 
-/ 1 +/ 2 — ^=^(^+4«^2), 
JlJi “ 
/l -/2 J (^ + 
—/ 1 —f ~ a ’ 
