16 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



Conversely, if a ^-zomal contain <f> 2 = by reason that it has two branches 

 each ideally containing <£ = 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 <£ = 0. 



35. Reverting to the case of the y-zomal curve 



Jl(& + Z<K) + Jm(e + M&) + &c. = , 



which does not contain $ = 0, each of the r(r — s) common points = 0, 

 (J> = is a 2 y - 2 -tuple point on the y-zomal; each of these counts therefore for 

 2 » ~ 2 intersections of the y-zomal with the curve 3> = 0, and we have thus the com- 

 plete number 2 v — 2 r (r — s) of intersections of the two curves, viz., the curve 

 $ = meets the »-zomal in the r (r — s) common points, each of them a 

 2 v - 2 -tuple point on the y-zomal, and in no other point. 



36. But if the v-zomal contains <£" = 0, then each of the r (r — s) common 

 points is still a 2 v_2 -tuple point on the aggregate curve; the aggregate curve 

 therefore passes 2 v ~ 2 times through each common point; but among these 

 passages are included w passages of the curve 4> = through the common point. 

 The residual curve— say the v-zomal — passes therefore only 2*— 2 — w times 

 through the common point ; that is, each of the r (?• — s) common points is a 

 (2»— 2 -co) tuple point on the p-zomal. The curve <J> = meets the y-zomal in 

 |2»— 2 r — a)(r — s)} {r — s) points, viz., these include the r{r — s) common 

 points, each of them a (2»— 2 — to) tuple point on the p-zomal, and therefore 

 counting together as (2*— 2 — w) r(r — s) intersections; there remain conse- 

 quently ws(r — s) other intersections of the curve = with the v-zomal. 



37. In the case where the v-zomal contains the factor q? u = 0, then throughout 

 excluding from consideration the r (r — s) common points = 0, $ = 0, the 

 remaining intersections of any zomal with its antizomal are points of contact of 

 the zomal with the v-zomal, and the remaining intersections of each pair of com- 

 plementary parazomals are nodes of the ^-zomal, it being understood that if any 

 zomal, antizomal, or parazomal contain a power of <I> = 0, such powers of 

 $ = are to be discarded, and only the residual curves attended to. The num- 

 ber 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 <£" = may present itself, 

 ideally in a single branch, or in several branches, and the consequent occur- 

 rence in the latter case of powers of $ = in certain of the zomals, anti- 

 zomals, or parazomals, the cases to be considered would be very numerous, and 

 there is no reason to believe that the results could be presented in any moderately 

 concise form ; I therefore abstain from entering on the question. 



