98 
ON A CERTAIN SEXTIC DEVELOPABLE 
[398 
substituting for 6 its value 
ST 
2 8’ 
and attending to the significations of S and T, we have 
2T8 X S-2S8 X T=0, 
ST8 y S-2S8 y T=0, 
ST8 Z S-2S8 Z T=0, 
ST8 W S — 2S8 W T= 0, 
which are in fact the conditions to be satisfied in order that the point (x, y, z, w) 
may belong to a nodal curve of the surface \/xu T- + 108 S 3 = 0. 
It is to be noticed that the coordinates of the before mentioned four points of 
intersection of the cuspidal and the nodal curves (being as already mentioned stationary 
points on the cuspidal curve) may be written x, y, z, w = (1, 1, 1, 1), (1, —1, — 1, 1), 
(-i, i, -i, a (-i. -i. 1.1). 
We have thus far considered the developable, or torse, the equation of which is 
{X' (x'w + y'z') + \j! (y'w 4- z'x) + v (z'w + x'y')} 3 
— 27V/aV (x'y'z + xy'w' + x z'w + y'z'w'f = 0, 
where V (V) + V (/1) + V (v) = 0 ; or, what is the same thing, writing a, b, c, in place 
of V (V), V (y!), \f {y') respectively, the torse 
{a 2 {xw + y'z') + b- {y'w' + z'x ) + c 2 iz'w' + x'y')} 3 - 27a-b 2 c- (x'y'z' + xy'w’ + x'z'w + y'zw')- = 0, 
where a + b + c = 0. 
Inverting this by the equations x, y', z, w' = -, - , -, - , we obtain a sextic 
CC XJ z \0 
surface 
{a 2 (xw + yz) + 6 2 (yiv + zx) + c- (zw + xy)) 3 - 27a 2 b-c- xyzw (x + y + z + w)* = 0, 
where a + 6 + c = 0; which surface I propose [secondly] to consider in the present paper. 
The surface has evidently the singular tangent planes x = 0, y- 0, z — 0, w = 0, 
each osculating the surface in a conic, that is, meeting it in the conic taken thrice, viz., 
x = 0, in a conic on the quadric cone a 2 yz + b 2 yw + c 2 zw = 0, 
y = 0, „ „ „ a 2 aw + b 2 zx + c 2 zw = 0, 
z = 0, „ „ „ a?xw + b 2 yw + &xy = 0, 
w = 0, „ „ „ o?yz + b 2 zx + c 3 xy = 0;