AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 145 
irt. 2. —To prove that the Locus of Conic Nodes of the Surfaces f(x, y, z, a) — 0 is 
a Curve, not a Surface. 
At every point of the locus of conic nodes the equations 
f(x,y,z,a) = 0.(13), 
D/ (x, y, z, a) _ 
Dx 
(14), 
D/Qe, y, X, a) 
Dy 
(15), 
D/ (x, y, z, a) 
Vz 
= 0 
(16), 
are simultaneously satisfied. 
In general these are satisfied by a finite number of values of x, y, z, a only. Hence 
there are only a finite number of conic nodes. 
The next case is that in which equations (13)-(16) are equivalent to three indepen¬ 
dent equations only, and then it is possible to satisfy them by relations of the form 
x=<f>(a), y — i a ), z = x (a ).(17). 
In this case there is a curve locus of conic nodes. But as such a locus is defined 
by two equations, it cannot be determined by equating a factor of the discriminant 
to zero. 
The next case is that in which equations (13)—(16) are equivalent to two indepen¬ 
dent equations only. Eliminating a between these, the equation of a surface is 
obtained. This is the case which will be further examined, and it will be shown that 
the tangent cone at every conic node must break up into two planes, i.e., the conic 
node becomes a binode. # 
Let £ rj, C be the conic node on the surface (10). 
Let £ -f- S£, r) + 8y, £ -f- Sf be the conic node on the consecutive surface 
J {x, y, z, ct + Sa) = 0.(18). 
Then 
/(£i?,O) = 0.(19), 
D/ (£ v> £ «) 
( 20 ), 
* In this connection may be noticed Art. 11, in which it is proved that if a surface have upon it a 
line at every point of which there is a conic node, then the tangent cone at every conic node must break 
up into two planes so that the line is a binodal line. 
MDCCCXCII.—A. U 
