949] 
on halphen’s characteristic n. 
469 
In fact, starting from an existing curve, say the complete intersection of two 
given surfaces of the orders ¡ju, v respectively, we have, as is known, 
d = /jLv, 2h = fiv (/jb — 1) (v — 1): 
we find also, as will appear, n = (/j, — 1) (v — 1): hence also nd = 2h, viz. the cone of 
the order n through the h nodal lines meets the cone of the order d in these lines 
counting each twice, and in no other lines. I remark also that h is = \n (n + 3) if 
¡i + v = 4 ; viz. we have here two nodal lines lying in a plane; but if /a + v > 4, then 
h is greater than \n (n + 3), viz. in this case the nodal lines are not h lines taken 
at pleasure, but they are lines subject to the condition of lying in a cone of the 
order n. But this is not all; the h nodal lines lie not only in this cone of lowest 
order n, but also in cones of the orders n+1, n+ 2, ..., n + (y, + v — 2) respectively: 
I do not for the moment attempt to determine the number of independent conditions 
which are hereby imposed upon the h lines. 
The last-mentioned theorem constitutes in fact the geometrical interpretation of 
results contained in Jacobi’s paper “ De eliminatione variabilis e duabus aequationibus 
algebraicis,” Grelle, t. xv. (1836), pp. 101—124, [Ges. Werke, t. ili. pp. 295—320]. 
Consider the two equations 
U = {*§%, y, z, wY = 0 ; V = (*$«, y, z, w) v = 0; 
representing surfaces of the orders y, v respectively; since the form of the equations 
is quite arbitrary, we may without loss of generality assume that the point 
(.x, y, z) = (0, 0, 0) 
is an arbitrary point in space; and this being so, we find the equation of the cone, 
vertex this point, which passes through the curve of intersection of the two surfaces 
by the mere elimination of w between the two equations. As the reasoning is exactly 
the same for a particular case, I write for convenience y = 3, v = 4, and consider the 
two equations 
AqU? + A-iVfi -)- A^w + A 3 = 0, 
I? 0 w 4 + B Y w z + B 2 w 2 + B 3 w + B 4 = 0, 
where the suffixes show the degrees in regard to (x, y, z), viz. A 0 , B 0 are mere 
constants, A 1} B } are linear functions (*$/», y, z) 1 , A.,, B 2 quadric functions (*$#, y, zf, 
and so on. Multiplying the first equation successively by 1, w, w 2 , w z , and the second 
equation successively by 1, w, w 2 , we have 7 equations from which to eliminate 1, 
w, w 2 , w z , w 4 , w 5 , w 6 , and the result is 
Ao, 
A, 
A, 
A 3 
A, 
A u 
A, 
A, 
^-Oj 
Aj, 
A 2 , 
A 3 , 
A 1} 
A 2) 
A 3 , 
Bo, 
B,, 
B 2 , 
B 3 , 
B 4 
B 0 , 
B 1} 
B 2 , 
B 3 , 
B 4 , 
B 1} 
B 2 , 
B 3 , 
Bi,