556 
ON POLYZOMAL CURVES. 
[414 
IV. VT+Vm + Vft + Vp = 0, a Vr+ 6 Vw + c Vw + c? Vp = 0; corresponds to IV. supra, 
viz., there is a branch ideally containing (z 2 = 0) the line infinity twice. But, observe 
that whereas in IV. supra, in order that this might be so, it was necessary to impose 
on l, m, n, p three conditions giving the definite systems of values VT : Vra : \/n : Vp 
= a : b : e : d, in the present case only two conditions are imposed, so that a single 
arbitrary parameter is left. 
V. VZ = Vm = Vw = v / p; corresponds to V. supra. 
VI. vT + Vm = 0, Vw + Vp = 0, a Vi + b sfm + c + d Vp = 0, or what is the 
same thing, vT : Vm : Vw : \/p = c — d : d — c : b — a : a — b\ the equation is thus 
(c — d!)(V2l 0 — V23°) — (a — b) (V2l° — V23°) = 0. There is here one branch ideally containing 
(z 2 = 0) the line infinity twice, and another branch ideally containing (z — 0) the line 
infinity once; order is = 5. Each of the points I, J is an ordinary point on the 
curve, the remaining points at infinity are a node (21° = 23°, (£° = 2)°), as presently 
mentioned, counting as three points, viz., one branch has for its tangent the line 
infinity, and the other branch has for its tangent a line perpendicular to the axis; 
or what is the same thing, there is a hyperbolic branch having an asymptote perpen 
dicular to the axis, and a parabolic branch ultimately perpendicular to the axis. The 
number of nodes is =5, viz., there is the node 21° = 23°, (£° = 3)° just referred to; and 
the two pairs of nodes ((c - d) V2P - (a - b) V(5 0 = 0, — (c — d) V23° + (a — b) V2)° = 0) and 
(c — d) V2P + (a — b) V2) 0 = 0, (c — d) V23° + (& — b) = 0), each pair symmetrically situate 
in regard to the axis. Hence also dps. = 5 ; class = 10 ; deficiency = 1. 
And there is apparently a seventh case, which, however, I exclude from the present 
investigation, viz., this would be if we had 
( 1 , 1 , 1 , 1 , )(VZ, Vm, \!n, Vp) = 0, 
a , b , c , d , 
a? , b 2 , c 2 , d 2 , 
a" 2 , b" 2 , c" 2 , d" 2 , 
that is, a, b, c, d denoting as before, if we had 
\Jl : y'm : \/n : \/p = a : b : c : d, and aa" 2 + b& ,/ ' 2 + cc" 2 + dd" 2 = 0. 
For observe that in this case we have 
a2P + b23° + c(S° + d3)° = 0, and ^ + ^ + ^+§ = 0; 
abed 
that is, the supposition in question belongs to the decomposable case. 
Article No. 197. The Decomposable Curve. 
197. We have next to consider the decomposable case, viz., when we have 
a2l° + b23° + c(£° + d2)° = 0;