ON POLYZOMAL CURVES. 
557 
4 14] 
see ante, Nos. 87 et seq.—it there appears that (unless the centres A, B, C, D are in 
a line) the condition signifies that the four circles have a common orthotomic circle; 
and when we have also 
l m n p „ 
- + t- + - + 3 = 0. 
abed 
The formulae for the decomposition are given ante, Nos. 42 et seq. Writing therein 
21°, 23°, (5°, iD° in place of U, V, W, T respectively, it thereby appears that the tetra 
zomal curve Vi2l° + Vm23° + \/n(i 0 + = 0, breaks up into the two trizomal curves 
where 
+ Vm$° + ^1° = 0, V42l° + VwU3° + Vw 2 (S 0 = 0, 
, a P_ 
+ d Vi’ 
and where we have 
A 
1 no 
> 
II 
a p 
+ d Vi’ 
A =' 
Vwj 
= V m 
“ \!bed 
£b 
1 
A, 
V nio = ' 
\A 
= A 
+ / \/bed 
p c 
1 c 
fm, 
A 2 = ' 
have 
L m, 
n, 
= 0, 
L m 2 
—f TT "• 
a b c 
a + b 
'm, 
Article Nos. 198 to 203. Gases of the Decomposable Curve, Centres not in a line. 
198. I assume, in the first instance, that the centres of the circles are not in a 
line; we have the following cases: 
I. No further relation between l, m, n, p; the order of the tetrazomal is = 8; 
the order of each of the trizomals is = 4, that is each of them is a bicircular quartic. 
II. v7 + Vra + Wn 4- fp = 0 ; the order of the tetrazomal is =7, that of one of 
the trizomals must be = 3. 
To verify this, observe that we have 
A 4- Vwj 4- = A 4- Vra + fn 4- (c fm — b Vn), 
or substituting for \!l 4- Vra + A the value — fp, this is 
Vi 
= |a fp — d A 4- sj^(cVm-b A)j, 
and similarly for V/ 2 + Vm 2 4- Vn 2 , the only change being in the sign of the radical 
/ad 
V be 
. But from the two conditions satisfied by l, m, n, p it is easy to deduce 
(a fp — d Vif —(c Vm — b Vnf = 0,