412] 
A MEMOIR ON CUBIC SURFACES. 
439 
171. Cuspidal curve. The equation of the surface may be written 
(x, — y, 3w\\2xw — y'\ 9zw + 8xy) 2 = 0, 
where 4x. 3w — y 2 = 12xiv — y 2 . This exhibits the cuspidal curve 12xw — y 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 
I2x, y, z 
y, w, — 8x 
Section XV = 12 - U 7 . 
Article Nos. 172 to 176. Equation WX 2 + XZ 2 + Y 2 Z = 0. 
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: p=b'= 1; t'= 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 .