AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
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(A.) The Discriminant. 
\y 2 — xy — mz (m — c) yz 
A = | — xy — mz x 2 (m + c) xz 
\(m — c) yz (m + c) xz (c 2 — g 2 ) z 2 — 2m xy z 
= mz 3 {2(/ 2 «y — (c 2 — e/ 2 ) mz}. 
The way in which the factor z 3 arises will now be examined. 
The discriminant is found by eliminating a, b between 
(bx — ay + cz) 2 — </ 2 z 2 — 2m (x — a) (y — b)z = 0 . . . . (a), 
— 2 y (bx — ay + cz) + 2m (y — b)z — 0 . . (ft), 
2x (bx — ay + cz) + 2m (x — a) z = 0 . (y). 
By means of (/3), (y), it follows that (a) can be written 
(bx — ay + cz) cz — # 2 z 2 — mz (2xy — bx — ay) = 0 . . . . (8). 
The values of a, b, satisfying (ft) and (y) are 
1 
— xy — mz (m — c) yz 
x 2 (m + c) xz 
(m — c) yz 
y~ 
Therefore 
(m + c)xz — xy — mz\ — xy — mz 
b 1 
xy — mz 
x 2 
■mxz {2 xy + (m + c) z} — myz {2 xy + (m — c) z} — mz (2 xy + mz) 
Now it will be shown that on the binode locus z = 0 ; therefore the values of a, b 
become indeterminate on the binode locus. 
But they may be determined by dividing out by the factor z, which vanishes on the 
binode locus, and then 
a = x(l ~b r—--\ 
\ 2 xy + mz) 
cz 
2 xy + mz 
Hence if y, 0 be any point on the binode locus, then at this point the values of 
the parameters are a = £, b = y. 
Hence there is a single set of values of the parameters satisfying the equations 
Df/Da = 0, Df/Db = 0 at points on the binocle locus, which has been determined. 
