26 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



ordinates 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 

 a; = 0, y = 0, 2 = 0, w = (viz., each of these is a double tangent of the curve), 

 and having besides the three nodes (x = y, z = w), (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 + fiy + yz + Sw = which is a double tangent of the curve; and writing 

 [7, V, W, T in place of x, y, z, w, 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 

 all + fiV + yW + ST = which is related to the curve in the same way with 

 the lines [7=0, V = 0, W = 0, T = ; or say there is not (besides the curves 

 [7=0, V=0, W = 0, T=0), any other zomal aU + /3V + y\V + ST = 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 a[7+/3V + yW + ST=0, 

 but belonging to a separate system. An instance of this will be mentioned in the 

 sequel. 



The Theorem of the Variable Zomal of a Trizomal Curve resumed — Art, Xos. 53 to 50. 



53. I resume the theorem of the variable zomal of the trizomal curve 

 JTU + JmV + JnW — 0. The variable zomal T = is the curve 



a£/+bF" + cW = 0, where a, b, c are connected bv the equation — l - j — — o : 



a b c ' 



that is, it belongs to a single series of curves selected in a certain manner out of 



the double series a U + bV + cW = (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 &U + bV + cW = 0, and of two 



of the three given curves, is identical with the Jacobian of the three given curves. 



I call to mind that, by the Jacobian of the curves U = 0, V = 0, W = 0, is 



meant the curve 



J(U,V,W)= d ^ V ' W) = 



d(x,y,z) 



d x U, d y U, d z U 

 d x V,d y V,a\V 

 d x W,d y W,d z W 



viz., the curve obtained by equating to zero the Jacobian or functional deter- 

 minant of the functions U, V, W. Some properties of the Jacobian, which are 

 material as to what follows, are mentioned in the Annex No. I. 



For the complete statement of the theorem of the variable zomal, it would 



be necessary to interpret geometrically the condition — \. j- + - = o, thereby 



a d c 



