AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 147 
If this were the only relation between these equations, they would determine the 
ratios 8 ^: By : 8 £ : 8 a. 
Hence there would be a curve locus, not a surface locus of conic nodes. 
If, then, there be a surface locus, equations (24)-(27) must be equivalent to two 
independent equations only. 
Expressing that (24)-(26) are equivalent to two independent equations only, it 
follows that 
p m m, cal 
D{£ v, ?} 
(29). 
But this is the condition that the tangent cone at the conic node should break up 
into two planes, and then the conic node becomes a binode. 
Hence there cannot be a surface locus of conic nodes, unless the conic nodes become 
binodes. 
Since equations (13)-(16) are equivalent to two independent equations only, every 
point on the intersection of the surfaces represented by (13) and (14) is a binode on 
the surface (13). 
Hence the surface (13) has a binodal line. 
The locus of these binodal lines is a surface at every point of which equations (l) 
and ( 2 ) are satisfied, hence it is a part of the locus of ultimate intersections, and its 
equation can be determined by equating a factor of the discriminant to zero. 
Art. 
3 .—To find the conditions which hold at every point on a Surface Locus of 
Binodcd Lines. 
In this case (29) holds. 
Hence, in order that (24)-(26) may give finite values for : S 17 : S£ : 8 a, 
p m ivi mi = 0 = p m w, hi 
pm [vi [?]} = 0 = p([a. [a. hi 
a {£?.«} 
p m> m> mi = 0 = p i m, [a, hi 
P {v, £ P {|> v, £} 
(30) , 
(31) , 
(32) . 
Now (30) shows that (27) depends on (24) and (25). Hence, in this case, the four 
equations (24-2 7) are equivalent to two independent equations only, which is obvious 
since (13-16) are equivalent to two independent equations only. 
u 2 
