563 
414] 
and then 
whence 
and we have thus 
and similarly 
ON POLYZOMAL CURVES. 
(d — a) Vi = (b — d) \/in + (c — d) Vn, 
(d — a) Vp = (a — b) Vra + (a — c) Vw, 
d*Jl — « Vw = ^—— (b Vw, —• c Vm), 
a — a 
'W— c 
Vra); 
^^+A=+\/ bt) < Wre ~ 0 Vm > : 
(observe that in the case not under consideration b Vn — c Vra = 0, and therefore 
VZj + V??^ + Vii 2 = 0, V/ 2 + Vra 2 + Vw 2 = 0). 
In the present case we have 
a: b : c: d = (6-c)(c-d)(d-6): - (c - d) (d - a) (a-c): (d-a)(a-b)(b-d) : -(a-b)(b-c)(c-a), 
and thence 
ad _(b — c) 2 
be (c£ — a) 2 ’ 
so that only one of the two sums Vi + Vra^ + V^, V^ + Vw 2 + V?? 2 is =0, viz., assuming 
/ad _ b — c 
V be d — a 
we have VZ X + V?7q + V?ij = 0. 
We have then also 
a^lx + b Vm x + c V?ij = a VZ + 6 Vra + c Vw 
+ 
Vp [ aa Vp .J± 
-\/ra (№,/s_ 
cc 
but we find 
and thence 
dd Vi — aa V« = ~ 7 —- C 
* a — a 
=- v/’ {a (di ' /l ~ oa ^ “ ■Vra (№ Vre " 
(6b Vw — cc V ra), 
cc 
a i Vm, + c VV, = ^ 
in virtue of a/= -y———• Hence V£ : Vra x : V^, = b-c : c - a : a-b, or the corre- 
V be d — a 
sponding trizomal is a conic, but the other trizomal is a quartic. 
71—2 
cc V ??i), = 0,