The collected mathematical papers of Arthur Cayley.

398] AND SEXTIC SURFACE CONNECTED THEREWITH. 97
and we have
X = 2 (/ - ) ( - ) (y - a) (a - ),
= 2 (y - 8) (a - ~) (a - /) (/- ),
= 2 (C - 8) (3 - y) (7y - a),
whence XLv = 8Xo /V.
Hence finally X, g, v denoting as just mentioned, and therefore satisfying
— + - =0, the equation of the developable is
Xuv {W3 - w (.2 + y2 + + ) + 2xyz}2 + 108 {(x + + v) ZU2 - X2 - y2 - v2}3 = O
(say this is XuvT2 + 108S=O0), and this surface (which has obviously the cuspidal curve
S = 0, T =0) has also the nodal curve
x2 (y2 - vzX) =y (z _ (X) - x y2)
/- y r — X VK — L
I will show a posteriori that this is actually a nodal curve on the surface. Introducing an arbitrary parameter 0, the equations of the curve may be written ut supra
Oxx = xw - yz,
Oyy = yw - zx,
Ovz = zw - xy,
20 (X + +v) w = 3w2 - y2- z
and we have thence, as before,
2w (3w — 3- ~ ^ = W _ S2 _ y _ z\
x y z
Hence
3w2 - x2 - y2 - z -2w (x" + y2 + z2)+ 6xyz
20=
(X + / + v) w _- Xx2 - zy2 _- z2
_3w - w (c2 + y2 + z2)
(X+ ^ + v) W2
3 {w3 - w (X2 + y + z2) + 2xyz}
(X + V + v) W2 - XX2 -. y2 - _z2 '
3T
S '
Hence writing
0(- 2Xx)-(-2wx + 2yz) =0,
(-2/y) - (- 2y + 2zx) = 0,
0 (-2vz) -(- 2wz + 2xy) = 0,
0.2(X ++ v)w-(3w2 - x2- -z2) = 0,

C. VI.

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