414] 
ON POLYZOMAL CURVES. 
485 
<£>“ = 0 may present itself, ideally in a single branch, or in several branches, and the 
consequent occurrence in the latter case of powers of 4> = 0 in certain of the zomals, 
antizomals, 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. 
Article Nos. 38 and 39. On the Trizomal Curve and the Tetrazomal Curve. 
38. The trizomal curve 
VF+Vf+VF=0 
has for its rationalised form of equation 
U 2 + F 2 + W 2 -2VW-2WC-2UV=0; 
or as this may also be written, 
(1, 1, 1, -1, -1, -1 )(U, V, Wf= 0; 
and we may from this rational equation verify the general results applicable to the 
case in hand, viz., that the trizomal is a curve of the order 2r, and that 
U = 0, at each of its r 2 intersections with F — W = 0, 
F=0, „ „ W-U = 0, 
W= 0, „ „ TJ-V= 0, 
respectively touch the trizomal. There are not, in general, any nodes or cusps, and 
the order being =2r, the class is = 2r(2r —1). 
39. The tetrazomal curve 
VF + VF+ VTF + Vr= 0 
has for its rationalised form of equation 
(F 2 + F 2 -f W' 2 + T' 2 — 1UV —2UW — ZUT— 2VW — 2FT — 2FT) 3 — 64>UVWT = 0, 
and we may hereby verify the fundamental properties, viz., that the tetrazomal is a 
curve of the order 4r, touched by each of the zomals TJ = 0, F = 0, W = 0, T= 0 in 
2r 2 points, viz., by U=0 at its intersections with \^U+^W + ^/T=0, that is, 
F 2 + IF 2 + T 2 — 2VW— 2VT — 2 WT=- 0 ; (and the like as regards the other zomals), and 
having 3r 2 nodes, viz., these are the intersections of (FF+v / F= 0, VlF + v / F=0), 
<VF+Vlf=o, \/v + 'Jt = 0), (VF+VF= o, Vf+V¥=o), or, what is the same thing, 
the intersections of (U — F — 0, W — T = 0), (U — W = 0, F— T =0), (U — T — 0, F— W = 0). 
There are not in general any cusps, and the class is thus = 4r (4r — 1) — 6r 2 , = 10r 2 — 4r.