12 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



is = (2 2 "- 5 — 2"~ 3 )r 2 , viz., this is = J 2"~ 2 (2"~ 2 — l)r 2 . But in this case the dps. 

 are nothing else than the r 2 common points, each of them a 2 v_2 -tuple point, the 

 y-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 

 9 = 0, <j> = 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 /0 + /'<£> = 0, mQ + m'& = 0, &c, and the equation of the v- zomal 

 curve is 



J I® + /'* + Jin® + ?/('* + &c. = 0. 



The rationalised equation is obviously an equation of the degree 2" -2 in 0, <j>, 

 giving therefore a constant value for the ratio 0: <£; calling this q. or writing 

 O = q$>, we have 



Jl<l + V + slmq + m' + &c. = , 



viz., the rationalised equation is an equation of the degree 2" -2 in q, and gives 

 therefore 2"~ 2 values of q. And the v-zomal curve thus breaks up into a system 

 of 2"~ 2 curves each of the form — q<$ = 0, that is, each of them in involution 

 with the curves = 0, <$> — 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. 



26. A more important case is when the zomal curves are each of them in 

 involution with the same two given curves, one of them of the order r, the other 

 of an inferior order. Let = be a curve of the order r, f = a curve of an 

 inferior order r — s; L = 0, M = 0, &c, curves of the order s j then the case in 

 question is when the zomal curves are of the form + L<$> = 0, + M$ = 0, 

 &c, the equation of the v- zomal is 



sll(Q + Z*) + Jr,t{® + M<s>) + &c. = 0, 



where /, m, &c, are constants. This is the most convenient form for the equation, 

 and by considering the functions X, M, &c. as containing implicitly the factors 

 l~\ m~\ &c. respectively, we may take it to include the form +/Iq + Z<J> 

 + s/mQ + M<& + &c. = 0, which last has the advantage of being immediately 

 applicable to the case where any one or more of the constants /, m, &c. may be = 0. 



27. In the case now under consideration we have the r(r — s) points of inter- 

 section ofythe curves = 0, $ = as common points of all the zomals. Hence, 

 putting in the foregoing formula k = r(r — 5), we have a v-zomal curve of the 

 order 2 v_2 r, having with each zomal 2 v ~ 2 rs contacts, or with all the zomals 

 2 v ~ 2 rsv contacts, having a node at each of the 2 v ~ i rs intersections (not being 

 common points = 0, $ = 0) of each pair of complementary parazomals ; that 

 is, together 2 v ~ i (2 v ~ l — v—\)rs nodes, and having, besides, at each of the 



