478 ON POLYZOMAL CURVES. [414 
16. Consider, secondly, a point for which fTJ+fV=0, VjP + VT = 0; to form the 
Norm, taking in this case the two factors 
vf+vf+vw+vt; 
Vh + VF- fW-fr, 
let their product 
= (V F + V F) 2 - (VF + VT) 2 
be called iT, and the product of the remaining six factors be called G; the rationalised 
equation is FG = 0, and the derived equation is FdG + GdF = 0. At the point in 
question G and dG are each of them finite (that is, they neither vanish nor become 
infinite), but we have 
F= 0, dF=(\/U+\'V)(dU+'/U + dV+\/V)-(>s/W + \/T)(dW+\'W+dT+*jT), = 0, 
that is, the derived equation becomes identically 0 = 0; the point in question is thus 
a singular point, and it is easy to see that it is in fact a node, or ordinary double 
point, on the tetrazomal curve. And similarly, each of the points of intersection of 
the two curves V U + V V = 0, fW+fT = 0 is a node on the tetrazomal curve. 
17. The proofs in the foregoing two examples respectively are quite general, and 
we may, in regard to a v-zomal curve, enunciate the results as follows, viz., in a 
y-zomal curve, the points situate simultaneously on two branches are either the inter 
sections of a zomal curve and its antizomal curve, or else they are the intersections 
of a pair of complementary parazomal curves. In the former case, the points in 
question are ordinary points on the y-zomal, but they are points of contact of the 
y-zomal with the zomal; it may be added, that the intersections of the zomal and 
antizomal, each reckoned twice, are all the intersections of the v-zomal and zomal. 
In the latter case, the points in question are nodes of the v-zomal; it may be added, 
that the y-zomal has not, in general, any nodes other than the points which are thus 
the intersections of a pair of complementary parazomals, and that it has not in general 
any cusps. 
Article Nos. 18 to 21. Singularities of a v-zomal Curve. 
18. It has been already shown that the order of the v-zomal curve is = 2 V ~ 2 r. 
Considering the case where v is =3 at least, the curve, as we have just seen, has 
contacts with each of the zomal curves, and it has also nodes. I proceed to determine 
the number of these contacts and nodes respectively. 
19. Consider first the zomal curve U = 0, and its antizomal V F + V W + &c. = 0, 
these are curves of the orders r and 2 V ~ 3 r respectively, and they intersect therefore 
in 2 V ~ 3 r 2 points. Hence the r-zomal touches the zomal in 2"~ 3 r 2 points, and reckoning 
each of these twice, the number of intersections is = 2 V ~ 2 r 2 , viz., these are all the 
intersections of the z^-zomal with the zomal U = 0. The number of contacts of the 
y-zomal with the several zomals U = 0, V = 0, &c., is of course = 2 V ~ 3 r 2 v.