414] 
ON POLYZOMAL CURVES. 
495 
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 £7=0, V = 0, W= 0, is 
meant the curve 
J(U, V, TF) = j( U '- V ’ F) = 
d(x , y , z ) 
d x U, 
dy £7, 
d z U 
d x V, 
d y V, 
d z V 
d x W, 
d y W, 
d z W 
viz., the curve obtained by equating to zero the Jacobian or functional determinant 
of the functions £7, 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 
- + + - = 0, thereby showing how 
cl D C 
the single series of the variable zomal is selected out of the double series of the 
curves a£7+bF+cTF=0 in involution with the given curves. Such a geometrical 
interpretation of the condition may be sought for as follows, but it is only in a 
particular case, as afterwards mentioned, that a convenient geometrical interpretation is 
thereby obtained. 
54. Consider the fixed line Sl=px + qy + rz = 0, and let it be proposed to find 
the locus of the (r — l) 2 poles of the line Sl = 0 in regard to the series of curves 
a,U + bV + cW = 0, where -+~ + - = 0. Take (x, y, z) as the coordinates of any one 
a d c 
of the poles in question, then in order that (x, y, z) may belong to one of the 
(r—l) 2 poles of the line XI =px + qy 4- rz = 0 in regard to the curve a[7 +bF +cTF= 0, 
we must have 
d x (aU+hV+cW) : d y (aU+bV + cW) : d z (aU+bV + cW)=p : q : r; 
or, what is the same thing, 
and these equations give without difficulty 
= d x Î2 : d v Sl : d z Sl ; 
a : b : c = J{V, W, il) : J(W, U, SI) : J (U, V, SI), 
whence, substituting in the equation 
l m . n _ , 
- + -z- + - = 0, we have 
a b c 
J(V, W, Sl) + ~J(W, U, Sl) + J(U, V, SI) 
as the locus of the (r—l) 2 poles in question. Each of the Jacobians is a function 
of the order 2r-2, and the order of the locus is thus = 4r - 4. As the given curves 
u = 0, V=0, W=0 belong to the single series of curves, it is clear that the locus 
passes through the 3 (r — l) 2 points which are the (r —l) 2 poles of the fixed line in 
regard to the curves £7=0, F = 0, W = 0 respectively.