AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 167 
(A.) The Discriminant. 
The discriminant is found by eliminating a between the above equation and 
3 [xjj ( x , y, z) — a]' 2 = 0. 
Hence it is [ (f> ( x , y, z )] 4 . 
Hence the locus of ultimate intersections is 
[<t>{x, y, Z )J = °- 
(B.) The locus of Unodal Lines which is also an Envelope is <f> (x, y, z) — 0. 
For, let g, y, C be any point on the surface <f> (x, y, z) = 0. Take a — \p(g, y, £), and 
consider the single surface 
But 
[(f) (x, y, z)J — [i ft (;x , y, z) — xjj (£ y, £) f = 0. 
x = g + X, y = y -j- Y, z = it, -f- Z. 
Then the lowest terms in X, Y, Z are 
X 3 4 + y/ 
Cp or] 
+ Z 
3 ? 
Hence g, y, £ is a unode on the surface 
[f (x, y, z)J - [i fj (x, y, z) - xfi (g, y. £)] H = 0. 
Hence the locus of unodal lines of the surfaces under consideration is (f) (x, y, z) — 0. 
Moreover, the uniplane 
is also a tangent plane to the locus of unodal lines. 
Hence the locus of unodal lines is also an envelope. 
Hence the factor [<f> (x, y, z)] 4 in the discriminant is accounted for. 
Section III. (Arts. 10-11).—Supplementary Remarks. 
Art. 10 .—Further remark on the case in which VhfgDag = 0. 
This condition indicates in general that the equation D/j/Dcq = 0 has two equal 
roots, but if/) be of the second degree in a } , D/^/Da, is of the first degree in cq, and 
hence it has either one root in a v or is satisfied by an infinite number of values of a v 
It is desirable to notice the latter case, because it corresponds to an important case 
treated in Part II., Section IV., of this paper. 
