* Ì 
[647 
line, and 
unaltered 
the fourth 
nt in place 
= 0. And 
rface K = 0 ; 
rface in this 
respectively; 
respectively; 
0; viz. this 
ermines the 
647] 
THE EQUATION, SYMMETRICAL DETERMINANT = 0. 
53 
conic in question, tut lor each of the planes K = 0, we have (a r , t/, c!’, f \ g\ h^&r, m, n)~ 
a peifeet squaie, or the conic a two-fold line; we have thus the 8 nodes lying in 
pairs on four lines, say the four “ rays, in the four planes K = 0 respectively; each 
of these rays meets the double line x = 0, y = 0 in a point; and we have thus on 
the double line 4 points, which are in fact pinch-points of the surface (as to this 
presently). It has just been stated that for the plane passing through the nodal line 
and a ray, the conic is a two-fold line (the ray twice) containing upon it a pair of 
nodes; more properly, the conic is the point-pair composed of the two nodes. 
We can find through the nodes four different plane-pairs; in fact, forming the 
equation 
- B + 2\F~ \-G = 0, 
this is 
l 2 (c + 2Xf + A 2 6) — 2l{g + A/i) (n + \m) + a {n + Am) 2 = 0; 
or, as this may also be written, 
[a(n + \m) — l (g + \h)f +l 2 (b — 2Af + A 2 c) = 0, 
where b, c, f and therefore also b — 2Af + \ 2 c are of the form (x, y) 1 ; say that we have 
b — 2\f + Arc ~(p, q, r§x, y) 2 , where p, q, r are of course quadric functions of A; 
determining A by the quartic equation pr — q 2 = 0, we have b — 2Af + A 2 c a perfect 
square, = {ax + /3y) 2 suppose; and we have thus the plane-pair 
[a (n + Am) — l (g + A/t)] 2 — l' 2 (ax + /3y) 2 = 0 
containing the eight nodes; viz. there are four such plane-pairs. The two planes of 
a plane-pair intersect in a line called an “axis”; that is, we have four axes each 
meeting the nodal line; and we have thus also through the nodal line and the four 
axes respectively four planes, which are “ pinch-planes ” of the quartic surface (as to 
this presently). 
It has just been seen that the equation B-2\F+C\ 2 = 0 (where A is arbitrary) 
is expressible in the form 
[a (n + Am) -1 (g + A/*)] 3 + l 2 (p, q, r^x, y) 2 = 0, 
viz. this is the equation of a quadric cone having its vertex on the nodal line at 
the point x = 0, y = 0, an — Ig + A {am -lh) = 0; this is, in fact, a cone touching the 
surface, as at once appears by writing the equation of the cone in the form 
hBC-F 2 + {\C-Fy} = 0, 
0 
that is, 
1>S7 + {\C-Ff}= 0: 
we thus see that, taking for vertex any point whatever on the nodal line, theie is a 
eircumscribed quadric cone.