EEA^. E. HAELET ON THE IMETHOD OE STMMETEIC PEODUCTS. 
331 
therefore 
- 4 / 5/1 
and 
consequently 
= ^(1, — 4 J, 4 ?aG, 1)^ 
= ^( 256 a^^— 64 a^ 5 (Z-hl 6 a 5 ^c— W") ; 
/;+/1+_^ = ^,{V^-‘iahtx+id‘%t) 
= ^( — 8 ae+ 35 ®); 
=l{( 8 «c- 35 ^)^-( 256 a^ 6 - 64 a^^ 6 ?+ 16 « 5 ^c- 35 ^)} ; 
or 
y^yi+/5-^+yi-^=7(-64«>«+16a“M+16«V-16«5V+35‘). 
^2 
It hence appears that/?, fl and -^2 are the roots of the cubic 
j\j 2 
(a^ 8a®c— 3a^5^ — 64«®e+16«'*JfZ-|-16«V— 16«5®<7+3i^ 1)®=0. 
When f is kno-uTi, x is given by 
and the corresponding formula) for .^2, ^’3, x^ may be readily obtained. 
6. Next, for the quintic 
{a, h, c, d, e, fj^x, 1)'=0, 
assume 
J Xi — h K^X 2 ~\~ ^'31^4“!" ^' 4'^55 
•N2 X^ ~j- ^’2.372“!“ kjlC^ “h ^l'^ 4 ~l“ ^ 3 '^ 5 ? 
X3 iTi-l- ^3>r'2“l“ k^X2~\' ^4^4“i“ 
X4 = .Tl ^ 4 ^* 2 -j- ^’35^3 + ^ 2'^4 “h 
and ^,(a:)=XiX2X3X,. 
In regard to the above distribution of the constants Jc^, k^-, h, ^4, it will be observed 
that they are armnged according to the following scheme : 
12 3 4 
2 4 13 
3 14 2 
4 3 2 1 
