414] 
ON POLYZOMAL CURVES. 
479 
20. Considering next a pair of complementary parazomal curves, an a-zomal and 
a /3-zomal respectively (a + /3 = v), these are of the orders 2 a ~ 2 r and 2^~ 2 r respectively, 
and they intersect therefore in 2 a+ ^~ 4 r 2 = 2 V ~ 4 r 2 points, nodes of the z7-zomal. This 
number is independent of the particular partition (a, /3), and the 7^-zomal has thus 
this same number, 2 V ~ 4 r 2 , of nodes in respect of each pair of complementary parazomals; 
hence the total number of nodes is = 2 V ~ 4 r 2 into the number of pairs of complementary 
parazomals. For the partition (a, /3) the number of pairs is = \y\ v -r- [a]“ [/3] 3 , or when 
a = /3, which of course implies v even, it is one-half of this; extending the summation 
from a = 2 to a = v — 2, each pair is obtained twice, and the number of pairs is thus 
= ^•2 {[i']" 4- [a] a [/3]^}; the sum extended from a = 0 to a = v is (1 +1) 1 ', = 2 V , but we 
thus include the terms 1, v, v, 1, which are together = 2v + 2, hence the correct value 
of the sum is = 2 V — 2v — 2, and the number of pairs is the half of this =2 V ~ X — v— 1. 
Hence the number of nodes of the y-zomal curve is = (2" -1 — v — 1) 2 V ~ 4 r 2 . 
21. The v-zomal is thus a curve of the order 2 v ~ 2 r, with (2 V 1 — v — l)2 v 4 r 2 
nodes, but without cusps; the class is therefore 
and the deficiency is 
_ 2>'-3 r [(v + 1) r — 2], 
= 2 V ~ 4 r [(v + 1) ?— 6] + 1. 
These are the general expressions, but even when the zomal curves U = 0, V=0, &c., 
are given, then writing the equation of the v-zomal under the form ^TÜ + \/mV+ &c. = 0, 
the constants l : m : &c., may be so determined as to give rise to nodes or cusps 
which do not occur in the general case ; the formulae will also undergo modification 
in the particular cases next referred to. 
Article Nos. 22 to 27. Special Case where all the Zomals have a Common Point or 
Points. 
22. Consider the case where the zomals JJ = 0, V= 0 have all of them any 
number, say k, of common intersections—these may be referred to simply as the common 
points. Each common point is a 2 , ' -2 -tuple point on the y-zomal curve; it is on each 
zomal an ordinary point, and on each antizomal a 2 l '~ 3 -tuple point, and on any a-zomal 
parazomal a 2 a_2 -tuple point. Hence, considering first the intersections of any zomal 
with its antizomal, the common point reckons as 2 V ~ Z intersections, and the k common 
points reckon as 2 v ~ 3 k intersections; the number of the remaining intersections is 
therefore = 2*'~ 3 (r 2 — k), and the zomal touches the y-zomal in each of these points. 
The intersections of the zomal with the i/-zomal are the ¿-common points, each of 
them a 2 , '“ 2 -tuple point on the i/-zomal, and therefore reckoning together as 2 v ~ 2 k 
intersections; and the 2" -3 (r 2 — k) points of contact, each reckoning twice, and therefore 
together as 2 v ~ 2 {r 2 — k) intersections (2 v ~ 2 k+ 2 v ~ 2 (r 2 — k) — 2 v ~ 2 r 2 , =r.2 v ~ 2 r)\ the total 
number of contacts with the zomals U= 0, V = 0, &c., is thus = 2 V ~ 3 (r 2 - k) v.