ON POLYZOMAL CURVES. 
554 
[414 
There is not, in general, any identical equation a2P + b33° + c(£° + d3)° = 0, but 
when such relation exists, and when we have also - + t- + - + ^ = 0, then the curve 
abed 
breaks up into two trizomals. When the conditions in question do not subsist, the 
curve is indecomposable. But there may exist between l, m, n, p relations in virtue 
of which a branch or branches ideally contain (z a = 0) the line infinity a certain number 
of times, and which thus cause a depression in the order of the curve. The several 
cases are as follows: 
Article No. 195. Cases of the Indecomposable Curve. 
195. I. The general case; l, m, n, p not subjected to any condition. The curve 
is here of the order =8; it has a quadruple point at each of the points I, J (and 
there is consequently no other point at infinity); it is touched four times by each of 
the circles A, B, C, D; and it has six nodes, viz., these are the intersections of the 
pairs of circles 
V^ + Vng 5 =0, fW +y/p%>° = 0, 
Vw6° +Vm° =0, Vmf + Vp$° = 0, 
fW + Vm23 o =0, VnG 3 +Vp2)° = 0; 
the number of dps. is 6 + 2.6, = 18, and there are no cusps, hence the class is = 20, 
and the deficiency is = 3. 
II. We may have 
vT+Vm + Vn+Vp = 0; 
there is in this case a single branch ideally containing (z = 0) the line infinity; the 
order is = 7. Each of the points I, J is a triple point, there is consequently one other 
point at infinity; viz., this is a real point, or the curve has a real asymptote. There 
are 6 nodes as before; dps. are 6 + 2.3, = 12; class = 18, deficiency = 3. 
III. We may have 
Cl + Vm = 0, Cn + Cp = 0; 
there are then two branches each ideally containing (z = 0) the line infinity; the order 
is =6. Each of the points I, J is a double point, and there are therefore two more 
points at infinity. These may be real or imaginary; viz., the curve may have 
(besides the asymptotes at /, J) two real or imaginary asymptotes. The circles 
VT2T+ CmM = 0, CnQi + Vp3) = 0, each contain (2 = 0) the line infinity, or they reduce 
themselves to two lines, so that in place of two nodes we have a single node at the 
intersection of these lines; number of nodes is =5. Hence dps. are 5 + 2.1, =7. Class 
= 16, deficiency =3. 
IV. We may have 
Cl : Vm : fn : Vp = a : b : c : d;