558 
ON POLYZOMAL CURVES. 
[414 
and hence one or other of the two functions 
+ Vwj + Vrq, VZ 2 + Vra 2 + V?2 2 is =0; 
that is, one of the trizomal curves is a cubic. 
III. Vz + Vp = o, Vm + = 0; order of the tetrazomal is = 6; and hence order 
of each of the trizomals is = 3. To verify this, observe that here 
Mr + i + « r + r =0, 
l ad 
which since a + b+ c + d = 0, gives — = ; so that, properly fixing the sign of the 
radical, we may write 
/7 / 
ite V Z + a / . 
m be 
Vm = 0. We have then 
V/j = —VZ, Vwq + VWj. 
abc 
(b + c) Vra; 
/ad 
which last equation, using ^ to denote as above, but properly selecting the signi 
fication of + , may be written 
, b + c 
Vwq + = + 
Vm. 
Hence 
VZj + (Vmj + Vwj) = ^ |(a + d) VZ + (b + c) nj^ Vmj 
=0, 
a + d f /-r 
viz., VZj + (V m 1 + V%) with a properly selected signification of the sign + is = 0; and 
similarly VZ 2 + (Vm 2 + Vw 2 ) with a properly selected signification of the sign + is =0; 
that is, each of the trizomals is a cubic. 
199. IV. VZ : Vra : Vw : Wp = a : b : c : d Rvalues which, be it observed, satisfy 
of themselves the above assumed equation - + ^ + -+ ^ = 0^; the order of the tetra- 
a D c a / 
zomal is =6; and the order of each of the trizomals is here again =3. We in fact 
have VZi = a + d, Vm x -f VVq = b + c, and therefore VZ X -f Vwj 4- = 0 ; and similarly 
VZ 2 +Vm 2 + Vw 2 = 0; that is, each of the trizomals is a cubic. 
I attend, in particular, to the case where the four circles reduce themselves to 
the points A, B, C, D; these four points are then in a circle; and the curve under 
consideration is 
a + b V33 + cV($-|-dV2)=0;