16 
It will be observed that 
x 0 -\-ax x + a?%2 — £ - 26 3 + 96c - 2 7d + 3 j (46 3 d 
i- 1 X. 
3 
y 2 
-imd+ic z + n<p I*] 
-(S + Hf, say. 
6 V 
and 
x 0 + a 2 <q + a # 2 = (S - H) 3 ; 
so that, since 
3<Tj — ■ 2<r “f* ci“(^ 0 Cq -f- ax x -f- cdx^) -t- ci(^Xq + ct x j + clx 
= - 6 + a 2 (S4-H) 3 +a(S-H)% 
and 
3^c 2 - 2# + a(x 0 + aXi + ct 2 x 2 ) + a 2 (x 0 + a 2 X x + ax 2 ) 
= - 6 + a(S + Hf + a 2 (S-H)% 
all the roots of the given cubic are included in the formula 
_ h + !«"'Y s + hV + L 2 ”'/ s + hV, 
where 
3 V / 3 V 
a — 3, 2, or 1. 
The Biquadratic. Let x 0 , x h x 2) x 3 be the roots of the bi- 
quadratic (or quartic) equation 
x 4 -}- bx 3 cx 2 + dx e~ 0 . 
Then 
2#+ (xq-\-x x — r 2 — x 3 )% + (x 0 — x x -\-x, — xfi -J- (x 0 — x x — x^-\-x^fi 
~^x-\- { (2<r) 2 — 4 2ir 0 a? 1 4 ( x Q x x -f- x 2 x z ) } * 
— j - 1 (2#?) 2 — 4 :]£jX(jX x -j— i(x Q x 2 -\-x x xi) ~\ z 
+ {(Sic) 2 — 42av*? 1 4-4(^ 0 ^3+<ri^ 2 )}^ 
= — 6+ (6 2 — 4c-f- 4q) - 4 - (6 2 — 4c4- 4£ 2 )*-f- (6 2 — 4c+4^) 4 , 
in which, 
/ ■' ■ ■ V) ry > L. /y» sv > / ■ — /%• ry > I _ y> ry \ / — — => /yi /y» ( /y* /W 
tq — ■■■ ■ <A/Quvj^ | tAy 2^3 5 (/2 ^0^2 I ^ 1^3 5 fc/g ■ ' ■ ' - cA/Qcvg | 
Now 
'2 i t—'ZxQX x =zc, 
^t x t. 2 —^xlx x x 2 —bd — 4c, 
t x t 2 t 3 —^xlx x x 2 x 3 -\- hxlx\x 
2 
2 — 
b 2 e — 4cc-f d 
) 
