442 A MEMOIR ON CUBIC SURFACES. [412 
178. The transversals are each facultative : p = b' = 3 ; t' = 1. 
179. Hessian surface. The equation is 
4>XYZW-(X+Y+Z + W)(WXY+ WXZ+ WYZ + XYZ) = 0, 
or, what is the same thing, 
X 2 (YZ + YW+ ZW) 
+ Y 2 (ZW + ZX + WX) 
+ £ 2 (WX + WY+XY) 
+ W 2 (X Y +XZ + YZ ) = 0. 
The complete intersection with the cubic surface is made up of the six axes each 
twice, and there is no spinode curve; a = 0, whence also ¡3' = 0. 
Reciprocal Surface. 
180. The equation is immediately obtained in the irrational form 
or rationalizing, it is 
f x + fy + ^ z -f f w = §, 
(x 2 + y 2 + z 2 + w 2 — 2yz — 2zx — 2xy — 2xw — 2yw — 2zui) 2 — Q4>xyzw = 0 ; 
so that this is in fact Steiner’s quartic surface. 
Nodal curve. This consists of the lines x — y = 0, z — w = 0; x — z = 0, y — iu = 0; 
x — w = 0, y — z = 0; so that b' = 3. 
To put any one of these, for instance the line x — y = 0, z — w = 0, in evidence, we 
may write the equation of the surface in the form 
[(¿c — y) 2 + (z — w) 2 -2(x-\- y)(z + £y)] 2 — 64xyzw = 0, 
that is 
{(¿c — y) 2 + (z — w) 2 } {(x — y) 2 + (z — w) 2 — 4 (x + y) (z + iy)} 
+ 4 [(,x + y) 2 (z + w) 2 — 1 Qxyzw\ = 0, 
or finally 
{(# - y) 2 + (z - w) 2 \ {O - y) 2 + (z - w) 2 -4 (x+y)(z + w)} 
+ 4 {(x — y) 2 (z — w) 2 + 4xy (z — w) 2 + 4ziu (x — y) 2 ) = 0, 
where each term is at least of the second order in x — y, z — w. 
There is no cuspidal curve; c' = 0.