PROFESSOR CAYLEY ON CUBIC SURFACES. 
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171. Cuspidal curve. The equation of the surface may be written 
(#, — y, 3w 1 ][12%w— y*, 9zw+ 8xy.y=0, 
where 4x . 3w— y 2 ='42xw — y 2 . This exhibits the cuspidal curve \2xw— # 2 =0, 
9zw-]-8xy=0, breaking up into the line w= 0, y=0 (reciprocal of edge) and a skew 
cubic; the line is really part of the cuspidal curve, or d—4. 
The equations of the cuspidal cubic may be written in the more complete form 
12x, y, 
y > w, 
Section XV=12-U 7 . 
Equation WX 2 +XZ 2 +Y 2 Z=0. Article Nos. 172 to 176. 
172. The diagram of the lines and planes is 
where the equations of the lines and planes are shown in the margins. 
173. The mere line is facultative: g'=b'= 1 ; if= 0. 
174. The Hessian surface is 
X 2 (XZ— Y 2 )=0, 
viz. this is the uniplane X=0 twice, and a quadric cone having its vertex at U 7 . 
The complete intersection with the surface is made up of X=0, Y=0 (torsal ray) 
