486 
ON POLYZOMAL CURVES. 
[414 
Article Nos. 40 and 41. On the Intersection of two v-Zomals having the same 
Zomal Curves. 
40. Without going into any detail, I may notice the question of the intersection of 
two v-zomals which have the same zomal curves—say the two trizomals f U+ VF+ V IF = 0, 
VH7+V m V+ fnW = 0, or two similarly related tetrazomals. For the trizomals, writing 
the equations under the form 
VZ7 + VF+VlF=0, flfU+fmfV+fnfW = 0, 
then, when these equations are considered as existing simultaneously, we may, without 
loss of generality, attribute to the radicals V LT, V F, V IF, the same values in the two 
equations respectively; but doing so, we must in the second equation successively 
attribute to all but one of the radicals fl, Vm, Vw, each of its two opposite values. 
For the intersections of the two curves we have thus 
VU : V F : V ÌF = Vm — Vw : Vn — fl : fl—fm, 
viz., this is one of a system of four equations, obtained from it by changes of sign, 
say in the radicals Vm and Vm Each of the four equations gives a set of r 2 points; 
we have thus the complete number, = 4r 2 , of the points of intersection of the two 
curves. 
41. But take, in like manner, two tetrazomal curves ; writing their equations in 
the form 
V77+ Vf+ Vtf+ Vt=o, 
flfU + Vm V F + fn V IF + Vp fT = 0, 
then VU, VF, VIF, VT may be considered as having the same values in the two 
equations respectively, but we must in the second equation attribute successively, say 
to Vm, Vw, Vp, each of their two opposite values. For the intersections of the two 
curves we have 
(Vm — fl) VF-h (Vn — Vz )VlF + (Vp — Vr ) ViT= 0, 
(VÌ-Vm)VF . + (Vw — Vm) fW + (Vp — Vm) fT = 0, 
viz., this is one of a system of eight similar pairs of equations, obtained therefrom by 
changes of sign of the radicals Vm, fn, Vp. The equations represent each of them a 
trizomal curve, of the order 2r; the two curves intersect therefore in 4r 2 points, and 
if each of these was a point of intersection of the two tetrazomals, we should have 
in all 8 x 4r 2 = 32r 2 intersections. But the tetrazomals are each of them a curve of 
the order 4r, and they intersect therefore in only 16r 2 points. The explanation is, 
that not all the 4r 2 points, but only 2r 2 of them are intersections of the tetrazomals. 
In fact, to find all the intersections of the two trizomals, it is necessary in their two 
equations to attribute opposite signs to one of the radicals V IF, ff; we obtain 2r 2 
intersections from the equations as they stand, the remaining 2r 2 intersections from the 
two equations after we have in the second equation reversed the sign, say of VT.