414] 
ON POLYZOMA.L CURVES. 
477 
V W, &c. to be those radicals which have the sign —. The foregoing equations break 
up into the more simple equations 
V U 4- &c. = 0, V W + &c. = 0, 
which are the equations of certain branches of the curves V U + &c. = 0, and V W + &c. = 0, 
respectively, and conversely each of the intersections of these two curves is a point 
situate simultaneously on some two branches of the original v-zomed curve V £/ + VF” +&c. = 0. 
Hence, partitioning in any manner the v-zome VI7-I-VF+&c. into an a-zome, V1/ = &c. 
and a /3-zome V W + &c. (a + /3 = v), and writing down the equations 
V U + &c. = 0, V W + &c. = 0 
of an a-zomal curve and a /3-zomal curve respectively, each of the intersections of 
these two curves is a point situate simultaneously on two branches of the v-zomal 
curve; and the points situate simultaneously on two branches of the v-zomal curve 
are the points of intersection of the several pairs of an a-zomal curve and a /3-zomal 
curve, which can be formed by any bipartition of the v-zome. 
14. There are two cases to be considered:—First, when the parts are 1, v — 1 (v — 1 is > 1, 
except in the case v = 2, which may be excluded from consideration), or say when the 
v-zome is partitioned into a zome and antizome. Secondly, when the parts a, /3, are 
each > 1 (this implies v = 4 at least), or say when the v-zome, is partitioned into a 
pair of complementary parazomes. 
15. To fix the ideas, take the tetrazomal curve Vi/ + VF+ V1T+ Vi 7 = 0, and 
consider first a point for which Vi/=0, VF+VTF+VT^O. The Norm is the product 
of (2 3 =) 8 factors; selecting hereout the factors 
VZ7+VF+ V1T + Vi 7 , 
VF- VF- Vf- Vt, 
let the product of these 
= U - (V F + V IF + VT) 2 
be called F, and the product of the remaining six factors be called G; the rationalised 
equation of the curve is therefore FG = 0. The derived equation is GdF + FdG = 0; 
at the point in question Vi/=0, Vp r +VTF-t-V2 1 = 0; G and dG are each of them 
finite (that is, they neither vanish nor become infinite), but we have 
F= 0, dF=dU-(^/V + \/W+'/T)(dV+'JV + dW+ \/W + dT+\/T), =dU, 
and the derived equation is thus GdU = 0, or simply dU = 0. It thus appears that 
the point in question is an ordinary point on the tetrazomal curve; and, further, that 
the tetrazomal curve is at this point touched by the zomal curve U = 0. And similarly, 
each of the points of intersection of the two curves V[/=0, VF+ VlF-f Vr = 0, is an 
ordinary point on the tetrazomal curve; and the tetrazomal curve is at each of these 
points touched by the zomal curve U = 0.