15 
1 7 1 
. . x x ~ - _ o + _ 
2 2 
and since 
2#! = 2a: — (# 0 — x^) ; 
'b 2 ~4cy. 
The Cubic. Let x 0 , x h x 2 be the roots of the equation. 
x z + bx* + cx + d = 0 , 
and let a be an unreal cube root of unity ; then 
3^0 — "SiX 4~ (Xq 4" ClXj_ 4" Cl x%) + (Xq + cl'“Xi H - ctX^) 
3 3. 
— ^jX + (oCq + o.x^ + cCx^j 4" (xq + cdx-^ + cix^j 
= %x 4- (Sr 3 + 6 x^X 0 , 4- Scl'Si'XqXi + 3a 2 S / ^o07 2 ) i 
4- (2<r 3 4- 6 x 0 x v x.> + 3a 2 S / ^Q.r 1 4- SaS'fl? 2 ^)** 
where, for shortness, I have introduced my cyclical symbol 
S'. (See Society’s Memoirs, second series, vol. xv., p. 185. 
See also Phil. Trans, for 1861, p. 333.) 
Now 
S'4ci+2 1 a9»* ~ = — (be — 3c?), 
and 
( /y& /y I I _ ry-Q /y> \ 
aj (A/Qf/t/g — ^(ri\w(r2 I ^i^o 1 / 
— /yi^/y* /y* /y>2 y»2/y,2\ 
— 1 tA/Qcv ^2 1 | - tA'QiAs'j^ ^ tft/Qbft/jxft/g / 
/v»4/y» /y> | V /yi^ /y>^ I. ^ /V^ svQ /y:2 
= ^_ 6M+c 3 +9 ^ 
Whence it appears that S 'x\xi and S'afe are the roots of 
the quadratic equation 
S /2 +(6c — 3^)2' -p& 3 d — 66cc?-|-c 3 -|-9<i 2 = 0 : 
so that, 
2' = — 1 (6c— 3d)± |(— 46 3 ^-p6 2 c 2 + 18 W— 4c 3 — 27d 2 )h 
Substituting one of these values for '2lx%x 1 , and the other for 
S'afe, in the above expression for Sx 0 , replacing the sym- 
metric functions of the roots by their values in terms of the 
coefficients, and reducing by aid of the known properties of 
a, we find 
3 & + g7f|. [ - 2& 3 + 9 be - 27 d + 3^3(4 b 3 d-bV - I8bcd 
2)H]i 
X 0 — — n b 4- 
+ 
+ 4c 3 + 27 d 
7 =[- 2& 3 + 96c - 27<7-3 j 3(4 b 3 d- bV-Ubcd 
+ 4c 3 + 27 d 1 ) j *]* 
