470 
ON HALPHEN’S CHARACTERISTIC n, IN THE 
[949 
viz. this is an equation 
(*$#, y, z) 12 — 0, 
of the cone of the order (¿=12, through the curve of intersection of the two surfaces: 
say this equation is fl = 0. 
But if we only multiply the first equation by 1, w, w 2 successively and the 
second equation by 1, w successively, then we have 5 equations serving to determine 
the ratios of w 5 , iA, w s , w 2 , w, 1, viz. we have these quantities proportional to the 
six determinants which can be formed out of the matrix 
A-o, 
A, 
A 2, 
A 3 
■A-0) 
A, 
A 2 , 
A 3 , 
A o, A 1} 
A 2 , 
A3, 
Bu 
B 2 , 
B 3 , 
B 4 
Bo, B 1} 
B 2 , 
B 3 , 
B4, 
say we have 
w 5 : w 4 
: w 3 
: w 2 
: w 
1 
= L : M 
: N 
: P 
: Q 
R, 
where L, M, N, P, Q, R represent homogeneous functions (*$#, y, zf, of the degrees 
11, 10, 9, 8, 7, 6 respectively. We may if we please write 
_I±_ M _N _P_Q 
W ~ M~ N~ P~ Q~ R’ 
or eliminating w, we have the series of equations which may be written 
L, M, N, P, Q 
M, N, P, Q, R 
viz. we thus denote that the determinants formed with any two columns of this 
matrix are severally = 0. This of course implies that each of the determinants in 
question is the product of il and a factor which is a homogeneous function of the 
proper degree in (x, y, z), so that the several equations are all of them satisfied if 
only i2 = 0. We have for instance PR — Q 2 — Ail, where A is a quadric function 
y, z) 2 ; similarly NR — PQ = BA, where B is a cubic function y, zf; and 
the like as regards the other determinants. 
If the ratios x : y : z have any given values such that we have for these 12 = 0, 
then iu has a determinate value, that is, on each line of the cone il = 0, there is 
a single point of the curve of intersection of the two surfaces: the only exceptions 
are when, for the given values of x : y : z, the expressions for w assume an inde 
terminate form, viz. w has then two values, and there are upon the line two points 
of the curve, or what is the same thing, the line is a nodal line of the cone: the 
conditions for a nodal line thus are L = 0, M = 0, N = 0, P = 0, Q = 0, R = 0, viz. 
each of these equations is that of a cone passing through the nodal lines of the 
cone il = 0; the cone of lowest order is R = 0, a cone of the order 6 meeting the