308 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
evidence, write for a moment x=y-\-u, z—y-\-y , then the equation is readily converted 
into 
w 3 . 1 6(« — y) 2 
+w 2 |— y(a 2 — 4«y+y 2 ) 1 
l + («+7)(2«— y)(-a+2y)J 
-f-wf y 2 (a 2 — 10ay+y 2 ) j 
j — 18 yoty(ct+y) l 
[-27 ay J 
+ y«y=0, 
which, each term being of the second order in a, y(=x—y, z—y ) respectively, exhibits 
the nodal line in question. 
164. Cuspidal curve. Multiplying by 27, the equation may be written 
(7y—3x—3z—5 i w, — ^+6w, — w^Sy+lbyw— \2x-\-zw-\-13w 2 , 
— 2§y 3 -\-24yx J r z—27xz— 8^w+16w 2 ) 2 =0, 
where 
4w {7y — 3x — 3z — bw) -f - ( — y + 6 w) 2 =y 2 + 1 3y w — 1 2{x + z)w + 1 6 iv 2 ; 
and we have thus in evidence the cuspidal curve, 
y 2 -{-16yw—12(x-\-z)w-{-18vf=0, 
—20y 2 -\-24y(x-\-z) — 27xz—8yw + 16w 2 =0, 
which is a complete intersection, 2x2, or quadriquadric curve; c'= 4. 
