PROFESSOR CAYLEY ON POLYZOMAL CURVES. 11 



Special Case where all the Zomals have a Common Point or Points — Art. Nos. 22 to 27. 



22. Consider the case where the zomals U — 0, V — 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"~ 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"- 3 intersections, and the k common points reckon as 2 V_3 k intersections ; the 

 number of the remaining intersections is therefore = 2" -3 (?- 2 — k), and the zomal 

 touches the y-zomal in each of these points. The intersections of the zomal with 

 the v-zomal are the k common points, each of them a 2"- 2 -tuple point on 

 the y-zomal, and therefore reckoning together as 2 v ~ 2 k intersections; and the 

 2 v-3 (r 2 — k) points of contact, each reckoning twice, and therefore together 

 as 2"~ 2 (> 2 — k) intersections {2"-' 2 k + 2"- 2 (r 2 — k) = 2"- 2 r 2 , = r . 2 v ~ 2 r) ■ 

 the total number of contacts with the zomals U = 0, V = 0, &c, is thus 

 = 2"- 3 (r 2 — k) v. 



23. Secondly, considering any pair of complementary parazomals, an a-zomal 

 and a /3-zomal, each of the common points, being a 2 a-2 -tuple point and a 

 2^~ 2 -tuple point on the two curves respectively, counts as 2 a + P~ i , = 2 V ~ A in- 

 tersections, and the k common points count as 2"- i k intersections; the number 

 of the remaining intersections is therefore = 2 y_4 (r 2 — ^), each of which is a 

 node on the y-zomal curve ; and we have thus in all 2" -4 (2 V_1 — v— 1) (r 2 — k) 

 nodes. 



24. There are, besides, the k common points, each of them a 2" -2 -tuple point 

 on the v-zomal, and therefore each reckoning as 12 V_2 (2''- 2 — 1), = 2 2v ~ 5 — 2"-- 

 double points, or together as (2 2 "- 5 — 2"- 3 )& double points. Reserving the term 

 nde for the above-mentioned nodes or proper double points, and considering, 

 therefore, the double points (dps.) as made up of the nodes and of the 2"- 2 -tuple 

 points, the total number of dps. is thus 



2"-*(2"- 1 -»- l)(r 2 -k) + (2 2y ~ 5 - 2"- 3 )k, 

 = 2"- i (2 v - 1 - »- l)r 2 + (/TT2"- 4 - 2"- 3 )A:; 



or finally this is 



_ 2" — *{ /2*"~ x — » — l)r 2 + (» — 1)1. 



so that there is a gain = 2 y_4 — 1)& in the number of dps. arising from the 

 k common points. There is, of course, in the class a diminution equal to twice 

 this number, or 2 v ~ 3 (v — l)k; and in the deficiency a diminution equal to this 

 number, or 2 v ~ i (v — 1) k . 



25. The zomal curves U — 0, V = 0, &c, may all of them pass through the 

 same v 2 points; we have then k = r 2 , and the expression for the number of dps 



