10 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



Singularities of a v-zomal Curve — Art. Nos. 18 to 21. 



18. It has been already shown that the order of the v-zomal curve is = 2 v --r. 

 Considering the case where v is = 3 at least, the curve, as we have just seen, has 

 contacts with each of the zomal curves, and it has also nodes. I proceed to deter- 

 mine the number of these contacts and nodes respectively. 



19. Consider first the zomal curve U — 0, and its antizomal J y + J W +&c. 

 = 0, these are curves of the orders r and 2 v ~ 3 r respectively, and they inter- 

 sect therefore in 2" _3 r 2 points. Hence the v-zomal touches the zomal in 2 v_3 r 2 

 points, and reckoning each of these twice, the number of intersections is = 2 y_ -'/•-, 

 viz., these are all the intersections of the j-zomal with the zomal U — 0. The 

 number of contacts of the v-zomal with the several zomals U = 0, V = 0, &c, is 

 of course = 2*~*r*p. 



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 ~*r and 2^- V 

 respectively, and they intersect therefore in 2 a + /3_4 r 2 = 2 p- *r a points, nodes of 

 the y-zomal. This number is independent of the particular partition (a, 6), and 

 the j'- 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"-*r 2 into 

 the number of pairs of complementary parazomals. For the partition (a, 6) the 

 number of pairs is = [y] w ■*■ [a]"[/3]0, or when a — /3, which of course implies v 

 even, it is one-half of this; extending the summation from a = 2to« = i/ — 2, each 

 pair is obtained twice, and the number of pairs is thus = ^2 \[y] v -5- [«] a [/^] /3 ( ; 

 the sum extended from a = to a = v is (1 + 1)", = 2", but we thus in- 

 clude the terms 1, v, v, 1, which are together = 2v + 2, hence the correct 

 value of the sum is = 2" — 2v — 2, and the number of pairs is the half 

 of this = 2 v ~ l — v — 1. Hence the number of nodes of the y-zomal curve is 

 = (2"- 1 — v — 1)2"-V. 



21. The v-zomal is thus a curve of the order 2"-' 2 r, with (2" _1 — v—l) 2 v ~ i r" 

 nodes, but without cusps ; the class is therefore 



= 2"~ 3 r[>+ l)r-2], 

 and the deficiency is 



= 2"- 4 r[(v + l)r- 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 

 JTU+ JmV + &c. = 0, the constants / : m : &c, may be so determined as to 

 give rise to nodes or cusps which do not occur in the general case ; the formula' 

 will also undergo modification in the particular cases next referred to. 



