90 
ON A CERTAIN SEXTIC DEVELOPABLE 
[398 
and thence in particular 
e — dXa + c^aß — b%aßy + a aßy8 = 0, 
viz. this is the linear relation which subsists identically between (a, b, c, d, e), the five 
linear functions of the coordinates (x, y, z, w). 
Starting from the above values of (a, b, c, d, e), we find without difficulty 
/ = (a — ft) 4 xy + (cl- y) 4 xz + (a — 8) 4 xw + (ß — y) 4 yz + (t3 — S) 4 yw + (7 — 8) 4 zw, 
J = (a — ß ) 2 (ß — 7) 2 (7 — a) 2 xyz + (a — ß) 2 (ß — 8) 2 (8 — a) 2 xyw 
+ (a - 7) 2 (7 - S) 2 (8 - a) 2 xzw + (ft - y) 2 (7 - 8) 2 (8 - ß) 2 yzw, 
but we thus see the convenience of introducing constant multipliers into the expressions 
of the four coordinates respectively, viz. writing 
X = (ßy 8) 2 x', 
y = (y8a) 2 y, 
z = (8aß f z, 
w = (aßy) 2 w, 
where for shortness 
(ßy8) = (ß-y)(y- 8) (8 - ß), &c., 
or what is the same thing, taking the quartic to be 
(a, b, c, d, e\t, l) 4 = x (¡3y8) 2 (t + a) 4 + y (78a) 2 (t + ft) 4 + z (8uft) 2 (t + 7) 4 + w' (afty) 2 (t + 8)*, 
we find 
J = (X'y'v') 2 (x'y'z' + yzw + zxw' + x'y'w), 
I = (X'/Mv) {V (xw' + yz) + y! (yw 4- z’x) 4- v (zw 4- x'y')}, 
where for shortness 
X' = (a - 8)2 (£ - 7 )2 } 
y' = (ft-8) 2 (7-a) 2 , 
v - (7 “ &Y (a ~ ¡3), 
or writing 
VM = (a —8) (ft- 7), 
V (p) = (ft- B)(7 - a), 
V(*/) = (7 - S) (a - /3), 
we have 
V (X) + V (p) + V ( v ) = 9, 
and the equation of the developable is thus 
{V(xw 4- y'z) 4- y!(y'w' 4- z'x) 4- v (xw + xy')} 3 — 27X'yv (xyz' 4- yzw + zxw' + x'y'w') 2 = 0. 
Observe that /=0 is a cubic surface passing through each edge of the tetra 
hedron, and having at each summit a conical point; I = 0 is a quadric surface passing 
through each summit of the tetrahedron, and at each of these points the tangent