494 
ON POLYZOMAL CURVES. 
[414 
equations 
give 
which are, as they should be, equivalent to two equations only, and which 
X : fi : v = 
1, 1, 1 
1, 1, 1 
1, 1, 1 
b, b', b" 
c, c', c" 
a, a', a" 
c, c , c" 
a, a, a' 
b, b', b" 
and the equation, with these values of X, ¡x, v substituted therein, is in fact the 
equation of the trizomal curve \/lU + VraF + VnW = 0 in terms of three new zomals. 
It is easy to return to the forms involving one new zomal and any two of the 
original three zomals. 
Article No. 52. Remark as to the Tetrazomal Curve. 
52. I return for a moment to the case of the tetrazomal curve, in order to show 
that there is not, in regard to it in general, any theorem such as that of the variable 
zomal. Considering the form flx + fmy + fnz + Vpw = 0 (the coordinates x, y, z, w are 
of course connected by a linear equation, but nothing turns upon this), the curve is 
here a quartic touched twice by each of the lines x=0, y — 0, z = 0, w- 0 (viz., each 
of these is a double tangent of the curve), and having besides the three nodes 
(x = y, z = vj), (x = z, y = w), (x — w, y = z). But a quartic curve with three nodes, or 
trinodal quartic, has only four double tangents—that is, besides the lines x = 0, y = 0, 
z = 0, w = 0, there is no line ax + ¡3y + yz + fav = 0 which is a double tangent of the 
curve; and writing U, V, W, T in place of x, y, z, iv, then if U, V, W, T are 
connected by a linear equation (and, a fortiori, if they are not so connected), there is 
not any curve aTJ + fiV + yW + hT = 0 which is related to the curve in the same way 
with the lines U = 0, V = 0, W = 0, T = 0; or say there is not (besides the curves 
17=0, V = 0, W — 0, T= 0), any other zomal aU +/3V+yW + 8T=0, of the tetrazomal 
curve. The proof does not show that for special forms of U, V, W, T there may 
not be zomals, not of the above form aU+¡3V+yW + ST = 0, but belonging to a 
separate system. An instance of this will be mentioned in the sequel. 
Article Nos. 53 to 56. The Theorem of the Variable Zomal of a Trizomal Curve resumed. 
53. I resume the foregoing theorem of the variable zomal of the trizomal curve 
VlU + VmV + VnW = 0. The variable zomal T=0 is the curve at7+bb r +cTB=0, where 
a, b, c are connected by the equation - + ^ + - = 0; that is, it belongs to a single 
3» D C 
series of curves selected in a certain manner out of the double series al7 + bF+clT=0 
(a double series, as containing the two variable parameters a : b : c). These are the 
whole series of curves in involution with the given curves U = 0, V= 0, W = 0, or being 
such that the Jacobian of any three of them is identical with the Jacobian of the three 
given curves; in particular, the Jacobian of any one of the curves ai7 + br+cW=0,