484 
ON POLYZOMAL CURVES. 
[414 
is, through a single branch ideally containing = 0), then the y-zomal will have two 
branches, each ideally containing = 0, and it will thus contain <T 2 = 0. In fact, if in 
the zomal and antizomal, or in the complementary parazomals, the branches which 
ideally contain = 0 are 
VZ(© + LW) + &c. = 0, Vw(© + N<&) + &c. = 0 
respectively (for a zomal, the + &c. should be omitted, and the first equation be written 
VT(® + L(p) = 0), then in the y-zomal there will be the two branches 
(Vl (®+ id>) + &c.) + (Vn(® + JV<P) + &c.) = 0, 
each ideally containing <I> = 0. 
Conversely, if a y-zomal contain d? 2 = 0 by reason that it has two branches each 
ideally containing d> = 0, then either a zomal and its antizomal will each of them, or 
else a pair of complementary parazomals will each of them, inseparably contain d> = 0. 
35. Reverting to the case of the z'-zomal curve 
Vl (® + LW) + Vm (©~+l/d>) + &c. = 0, 
which does not contain = 0, each of the r(r— s) common points © = 0, <J> = 0 is a 
2 l ' _2 -tuple point on the y-zomal; each of these counts therefore for 2 V ~ 2 intersections 
of the v-zomal with the curve d> = 0, and we have thus the complete number 
2 V ~ 2 r 0 f intersections of the two curves, viz., the curve <f> = 0 meets the y-zomal 
in the r (r — s) common points, each of them a 2" _2 -tuple point on the v-zomal, and 
in no other point. 
36. But if the I'-zomal contains = 0, then each of the r (r — s) common points 
is still a 2 , ' _2 -tuple point on the aggregate curve; the aggregate curve therefore 
passes 2 1 ' -2 times through each common point; but among these passages are included 
w passages of the curve d> = 0 through the common point. The residual curve—say 
the y-zomal—passes therefore only 2 U ~ 2 — w times through the common point; that is, 
each of the r (r — s) common points is a (2" _2 — <w) tuple point on the z/-zomal The 
curve d> = 0 meets the y-zomal in {2 V ~ 2 r — co (o— s)} (r — s) points, viz., these include 
the r(r — s) common points, each of them a (2 V ~ 2 — co) tuple point on the v-zomal, and 
therefore counting together as (2 V ~ 2 — &>) r (r — s) intersections; there remain consequently 
co s (r — s) other intersections of the curve ® = 0 with the v-zomal. 
37. In the case where the r-zomal contains the factor d>“ = 0, then throughout 
excluding from consideration the r(r — s) common points © = 0, d> = 0, the remaining 
intersections of any zomal with its antizomal are points of contact of the zomal with 
the I'-zomal, and the remaining intersections of each pair of complementary parazomals 
are nodes of the r'-zomal, it being understood that if any zomal, antizomal, or parazomal 
contain a power of d> = 0, such powers of d> = 0 are to be discarded, and onlv the 
residual curves attended to. The number of contacts and of nodes may in any 
particular case be investigated without difficulty, and some instances will present 
themselves in the sequel, but on account of the different ways in which the factor