AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 149 
in order that (27) may be satisfied, 
[a, a] = 0.. . . (40). 
And now (27) becomes an identical equation. It does not depend on (24). 
Art. 5.— Examination of the Discriminant A, and its Differential Coefficients, when a 
Surface Locus of Binodal Lines exists. Proof that A contains B 2 as a factor. 
Let 77 , £ be a point on the binodal line on the surface ( 10 ). 
Then when x = £ y = 77 , z = £, 
/i = 0 , 
Hence by ( 6 ) and (7), when x = f y — 77 , z = £, 
Similarly 
A = 0 , |= 0 . 
rv - V, ^ - U. 
oy 02 
Hence if B = 0 be the equation of the surface locus of binodal lines, A contains B 2 
as a factor. 
Example 2 .— Locus of Binodcd Lines. 
Let the surfaces be 
[(/> (x, y, z) - af + x (x, y, *) j> (d y, 2)] 2 = 0 . 
A. The Discriminant. 
This is found bv eliminating a between the above, and 
— [2 <t> (x> y, z) — a] = 0. 
Hence the discriminant is y {%> y > 2 ) [i/f (x, y, z)f. 
Hence the locus of ultimate intersections is 
X (x, y, z) [tf, (x, y, z)f — 0. 
B. The Locus of Binodal Lines is xp (x, y, z) = 0 . 
Let £, 77 , £ be any point on the surface xp (x, y, z) = 0 , 
