17 
so that the three values of t are given by the solution of the 
cubic equation 
?—c?+ (bd+-±e)t—-(b 2 e—±ce+d 2 ).=:0, 
and are all therefore included in the formula 
— ic+ia’"(S+ H )* + | 
where a = 3, 2 or 1, and is an unreal cube root of unity, 
and 
S=l(27b 2 e—%cd+2<?—72ce+21d?y 
H— 3 V 3 /276V j — 1 86 3 cde + 46 3 rf 3 +46V« — bVcP 
— 1 4Ab 2 ce 2 + 6 V 2 d 2 e +80 bc 2 de — 1 8 6cd 3 + 192 bde 2 
—1 6c i e+4cW+ 1 28cV— 144«2 2 e+27<2 4 — 256e 3 V 
We have then 
**=- i^+i { 6 -T c +K S+H ) 5+ K S-H ) i }* 
+1 { *»-V 6 «+3-(s + H )^.(8- H )*j* 
+2 { 6 ~T C+ S a ( s+H )* +5“ 2 4 
and if, for shortness, we write 
x 0 =i(_6 + T 5 +Ts i +T ^, 
the remaining roots of the quartic will be given by the for- 
mulae 
= i(-6+T*— T,*— 1 
x*=\{-b- T,4 T 2 J — T, 4 ), 
- i(— 6— T, 4 — TZ+Ts*). 
Here however it is important to notice that since 
T/T 2 2 To (x 0 -\-x x — x 2 — x z ) (x 0 — x x -j-x 2 — x 3 ) (x 0 — x x — x 2 +x 3 ) 
6 3 +4 bc—8d, 
the above solution only applies when — 6 3 +46c — 8 d is posi- 
