552 
ON POLYZOMAL CURVES. 
[414 
and the equations - + ^ + - = 0, \Zl + \Zm + */n = 0, then give for \fl, Vra, Vrc two sets 
Sj DC 
of values, viz., these are 
and 
\/l : Vm : \/n = b 3 — c 3 : c s — a 3 : a 3 — b 3 , 
— b 3 + c 3 : — c 3 — a 3 : a 3 -b 3 \ 
and we again obtain the equations of the two cubics, each equation under a fourfold 
form, viz., these are 
and 
( • , 
— c 3 + d 3 , — 
d 3 + b 3 , 
o 3 b 3 
) (VS1, VS3, Vg, V$) = 0, 
— d 3 + c 3 , 
• J 
a 3 + d 3 , 
a 3 -c 3 
— b 3 + d 3 , 
- d 3 + a 3 , 
b 3 —— a 3 
b 3 c 3 , 
c 3 ct 3 , 
a 3 -b 3 , 
• 
( • , 
c 3 d 3 , 
d 3 + b 3 , 
C3 b 3 
)(Vsf, VS3, Vg, V$) = 0. 
d 3 C3, 
• > 
«3 C?3» 
a 3 + c 3 
1 
rO 
1 
c? 3 + a 3 , 
• > 
63 - a 3 
b 3 + c 3 , 
C3 Cl 3 , 
a 3 ~ b 3 , 
• 
The three 
systems have 
been 
obtained independently, but they 
course be derived each from any other of them: to show how this is, recollecting that 
we have 
RA, RB, RG, RD = a 1} b u c lf d 1} 
SA, SB, SC, SD = a 2t b 2 , -c 2 , -cl 2 , 
TA, TB, TC, TD = a 3y b 3 , c 3 , d 3 ; 
then to compare 
the similar triangles 
and the similar triangles 
(a 1} &!, Cj, dA and (a 2 , b 2 , c 2 , d 2 ) ; 
SBC give — Cj : — c 2 : b 2 , 
SAD =a 1 —d 1 : —d 2 : a 2 , 
RA G give a 2 — c 2 
RBD = 6., — d 9 
C\ d\, 
d\ ‘ ¿i ; 
using these equations to determine the ratios of a 2 , b 2 , c 2 , d 2 we have 
that is 
, or ^«2 — C^Ca — cA + cA = 0 ;