436] 
ON A CERTAIN SEXTIC TORSE. 
101 
Standard Equation of the Unicursal Quartic. 
2. The coordinates (x, y, z, w) being originally taken to be proportional to any 
four given quartic functions (*fd, l) 4 of the parameter 6, then forming a linear 
function of the coordinates, we have four sets of values of the multipliers, each reducing 
the function of 6 to a perfect fourth power; that is, writing (X, F, Z, W) for the 
linear functions of the original coordinates, and taking (X, F, Z, W) as coordinates, it 
appears that the unicursal quartic may be represented by the equations 
X : F 
: Z : W- 
= (0 + a) 4 : 
(0 + pY ^ 
(0 + y) 4 
(0 + 8y. 
Tangent Line, and Osculating Plane of the Unicursal Quartic. 
3. The 
equations of the tangent line 
at the point (0) 
(that is, the point 
COOl 
dinates whereof are as (0 + a) 4 
: (0 + Æ) 4 
: (0 + 7) 4 
(«+*)') 
are at once seen to 
x, 
F, 
F, 
w 
= 0, 
(0 + a) 4 , 
{0 + PY. 
(0 + y) 4 > 
(0 + 8y 
(0 + a) 8 , 
(0+Æ) 3 , 
(0 + 7) 3 , 
(0 + 8y 
and 
that of the osculating plane to be 
X, 
F, 
W 
= 0. 
(0 + a) 4 , 
(6 + p Y, 
(0 + y) 4 > 
(0 + S) 4 
(0+a) 8 , 
(0+ /3) 8 , 
(0 + 7) 3 , 
(0+8) 3 
(0 + a) 2 , 
(0+Æ) 2 , 
(0 + vY, 
(0 + 8) 2 
Writing 
as in the sequel 
a = /3 — 7, 
f=a-B 
0 = 7 — a, 
g = P-S 
c = a - /3, 
/i = 7 — 8 
the 
equations 
of the tangent line become 
hY 
(0 + /3) 8 
, 
(0 + y) 3 
aZ 
(0 + S) 3 — 
0, 
hX 
(0 + a) 3 
+ 
fZ 
(0 + *iY 
hZ 
(0 + sy 
0, 
gx 
/F 
+ 
cZ 
0, 
(0+ a) 8 
(0 + /3) 3 
(0 + 8f 
aX 
hY 
cZ 
0, 
(0 + a) 8 
(0 + /3) 3 
(0 + y) 3 
(equivalent of course to two equations), and the equation of the osculating plane becomes 
ahgX hh/Y 
gfcZ abcW _ Q