The collected mathematical papers of Arthur Cayley.

412] A MEMOIR ON CUBIC SURFACES. 421
or substituting for /3, y, 8 in the first and last lines their values
(= + z +, xz + xw + zw, xzw),
this is
4a3y2 (y - x) (y - z) (y - w)
+ a2 {(12y - /32) y4 - (8/3 + 368) y3 + (:30o8 + 872) y - 3678y + 2782}
+ 2a {(6,2 _ /3, - 9/38) y2 + (12/328 '- 2/y2 - 18y) y + 273 + 2782 - 9/378}
( - Z)2 ( - ) () - 2(z )2 = 0.
123. The nodal curve is made up of the lines (y=x=z), (y=x=w), (y=z=w),
reciprocals of the three transversals.
To show this I remark that, writing
3' = ( - ) + ( - y) + (w - y),
' = (x - y) (z - y) + (x - y) (w - y) + (z - ) (w -),
8' = (X-y)( -y)(w-y),
the equation of the surface may be written
4aly2 (y - x) (y - z) (y - w)
+ a2 {y2 (12/'3' - y/2) + x. 187'8' + 278'2}
+ 2a {y (- 6/3'2' + 2/'y/2 + 9y'8') + 2/y'3 + 278'2 - 9/'y8'}
-(x - z)2 (x - W)2 (z - w)2 = 0,
whence observing that y' is of the order I and 8' of the order 2 in (x -y), (z -y)
conjointly, each term of the equation is at least of the second order in (x-y), (z-y)
conjointly; or we have y=x=z, a nodal line; and similarly the other two lines are
nodal lines.
124. The foregoing transformed equation is most readily obtained by reverting to
the cubic in T, U, viz. writing p=x-y, r =z-y, s=w-y, and therefore x=y + p,
= y + r, w=y+s, the cubic function (putting therein T= V+ yU) becomes
a(V+ yU)V ) V ( (V-rU)(V- sU);
writing /3', y', 8' = p + r + s, pr + ps + rs, prs, the coefficients are (3 (a + 1), ay - /3', 7, - 38'),
and the equation of the surface is thus obtained in the form
27 (a + 1)2 8/2
+ 18 (a + 1) (ay - ') y'8'
+ 4(a+ 1)7'3
- 4 (ay-/')3 8'
- (ay- _/)2'2 = 0,
which, arranging in powers of a, and reversing the sign, is the foregoing transformed
result.