AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 219 
Representing both values of Y by Y = XX, it follows that 
aX 3 + 2c (c + eX) XZ + 2 cgzZ 2 = 0. 
Therefore, 
X/Z = - - (c + eX) ± - [cHc + eX) 3 - 2 c*qz} v \ 
Hence, 
x — a — — - (c +cX) ± ~ (c 3 (c + eX ) 3 — 2cagz } y2 .... (13G), 
y-b = X j—^(c+cX)dh;{c 3 (c + cX) 3 -2c« 9 z} 1/3 j . . . (137). 
These give the values of a, b which, when substituted in the equation of the 
surfaces, give the discriminant. 
The values of a, b corresponding to a point £, rj, £ on the binode locus, will now be 
found. 
It will be shown presently that z = 0 is the binocle locus. 
Hence £ = 0 , and therefore 
£ — a — — ~~ (c + eX) dz " (c + eX), 
CL CL 
V - h = X j- ^ (c+ eX) ± ^(c + eX)j. 
Hence, for each value of X, one of the values of a is £, and one of the values of 
b is 7]. 
Hence there are two sets of values of a, b satisfying Df/Da = 0 , Df/Db = 0 , which 
become equal when x = y = 77 , 2 = 0 . 
These two sets of values both give a — b = 17. 
It will now be shown that the substitution of each of these systems of values of 
a, b in /, will give rise to the factor z 3 in A. 
Now 
Y = XX, X = pZ, 
where 
X = ± \/{ea/cf3), 
g = — (c + eX) d= ~ {c 3 (c d - eX ) 2 — 2cagz} 1/2 . 
Substituting these in the left-hand side of (133), it becomes 
[(cp d~ eXp -T ( 72 ) (cp d- cXp -f 3 ^ 2 ) -- AV] Z 3 , 
i.e., 
[p 3 (c d~ eXf + 4yp (c + eX) z + (3/ — A 3 ) 2 3 ] Z 3 . 
2 f 2 
