480 
ON POLYZOMAL CURVES. 
[414 
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 3 ~ 2 -tuple point 
on the two carves respectively, counts as 2 a+ P~ 4 , = 2 V ~ 4 intersections, and the k common 
points count as 2 v ~ 4 k intersections; the number of the remaining intersections is there 
fore = 2 V ~ 4 (r 2 — k), each of which is a node on the I'-zomal curve; and we have thus 
in all 2 V ~ 4 (2 V ~ 4 — v — 1) (r 2 — k) nodes. 
24. There are, besides, the k common points, each of them a 2" -2 -tuple point 
on the z'-zomal, and therefore each reckoning as \2 V ~ 2 (2 V ~ 2 — 1), = 2 2u ~ 5 — 2 V ~ 3 double 
points, or together as (2 2l/ ~ 5 — 2 V ~ 3 ) k double points. Reserving the term node 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 v ~ i (2 v ~ 1 -v-l)(r 2 -k) + (2 2v ~ 5 — 2" -3 )k, 
= 2 V ~ 4 {2 v ~ l - V - 1) r 2 + {(i; + 1) 2 V ~ 4 - 2 V ~ 3 } k; 
or finally this is 
= 2 P_4 {(2 p - 1 - v - 1) r 2 + — 1)} ; 
so that there is a gain = 2 V ~ 4 (y — 1) k 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 (y —1)&; and in the deficiency a diminution equal to this number, or 
2 V ~~ 4 (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. is 
= (2 2v ~ 5 — 2 v ~ 3 )r 2 , viz., this is =\2 V ~ 2 (2 V ~ 2 — \)r 2 . But in this case the dps. are nothing 
else than the r 2 common points, each of them a 2 p ~ 2 -tuple point, the i/-zomal curve 
in fact breaking up into a system of 2 V ~ 2 curves of the order r, each passing through 
the r 2 common points. This is easily verified, for if © = 0, O = 0 are some two curves 
of the order r, then, in the present case, the zomal curves are curves in involution 
with these curves ; that is, ’ they are curves of the form IS + l'<i> = 0, m© + m ,( t> = 0, &c., 
and the equation of the v-zomal curve is 
V/© + ¿'<l> + Vm© + m'd? + &c. = 0. 
The rationalised equation is obviously an equation of the degree 2 V ~ 2 in ©, T>, giving 
therefore a constant value for the ratio © : ; calling this q, or writing © = q<t>, we 
have 
\/lq + V + mq + rri + &c. = 0, 
viz., the rationalised equation is an equation of the degree 2 V ~ 2 in q, and gives there 
fore 2 V ~ 2 values of q. And the i/-zomal curve thus breaks up into a system of 2 V ~ 2 
curves each of the form © — qQ? = 0, that is, each of them in involution with the 
curves © = 0, <f> = 0. The equation in q may have a multiple root or roots, and the 
system of curves so contain repetitions of the same curve or curves; an instance of 
this (in relation to the trizomal curve) will present itself in the sequel; but I do not 
at present stop to consider the question.