414] 
ON POLYZOMAL CUPVES. 
565 
YII. If we have 
1 , 
1,1, 
1 
= 0, and ( 
1 , 
1 , 
1,1) 
a , 
b , c , 
d 
a 
b , 
c , d 
a 2 , 
b 2 , c 2 , 
d 2 
a 2 , 
b\ 
c 2 , d 2 
a" 2 , 
b" 2 , c" 2 , 
d" 2 
b" 2 , 
c" 2 , d" 2 
and thence 
the tetrazomal has a branch ideally containing (z 3 = 0) the line infinity 3 times; order 
is = 5; orders of the trizomals are 3, 2. We have here 
VZ : Vm : fn : Vp = a : b : c : d, 
VZ^ = a + d , V^=a + d , 
V?^ = b Vm 2 = b 4- 
/— /bed /— /bed 
A =c +A / a , =c-/ V / —. 
VZ] + Vm! + V% 1 = 0, \A 4-Vm 2 + V/i 2 = 0. 
a*fli + b Vwj H-cVwi = a (a + d) +Z>b + cc 
= ( a -d)d-(6-c) / / b “- 
= d|(a-d)-(6-c)^/^|, 
which give 
Moreover 
and similarly 
whence in virtue of 
fL 4- b Vm 2 4- c = d j(a — d) 4- (6 — c) 
ad _ (b — cf 
be (d - a) 2 ’ 
one of the two expressions is = 0; and the trizomals are thus a conic and a cubic. 
Article No. 206. The Decomposable Carve; Transformation to a different set of 
Concyclic Foci. 
206. Consider the decomposable case of 
VZ2( + Vra33 4- V?i(S 4- = 0 ; 
^ . . . - . I m n p _ 
viz., the points A, B, C, D lie here in a circle, and we have -+^ + - + ^ — 0. 
Taking (A lt A) the antipoints of (A, D); (BA) the antipoints of (B, C)\ then