[436 
436] 
ON A CERTAIN SEXTIC TORSE. 
103 
equations which give 
3uartic. 
3S (sc, y, z, w) 
h - g + a 
-h . +f +b 
9 ~f • +c 
— a —b — c . 
= 0, 
= 0, 
= 0, 
= 0, 
h/3 — gy+a8 = 0, 
— ha . -(- fy -f- b8 — 0, 
ga —f/3 . + cS = 0, 
— aa — b/3 — ay . =0, 
and also 
af+ bg + ch — 0, 
then the discriminant is a function of (x, y, z, w), (a, b, c, f g, li) of the degree 10 
in (x, y, z, w) and the degree 30 in (a, b, c, f, g, li). But the equation in 6 has two 
equal roots, or the discriminant vanishes, if any one of the quantities (x, y, z, tu) is 
= 0; and again, if any one of the differences a — /3, &c. (that is any one of the 
quantities a, b, c, f, g, h) is = 0: the discriminant thus contains the factors xyzw and 
(■abcfghy, and throwing these out, we have an equation of the form 
A = (a, b, c, f, g, A) 18 (x, y, z, wf = 0, 
which is the equation of the sextic torse. 
Principal Sections of the Torse. 
6. Consider for instance the section by the plane w = 0. Writing w = 0, the equation 
of the osculating plane is 
(0 + [sc (6 + /3) 2 (6 + 7 ) 2 + y (0 + y) 2 (0 + <*)'- + * (6 + af (0 + £) 2 ] = 0. 
The discriminant of the sextic function vanishes identically in virtue of the double 
. . factor (0 + 8) 2 . But omitting this factor, the equation becomes 
,he form 
x (0 + /3) 2 (0 + y)’ 2 + y (0 + y) 2 (0 + a) 2 4-z(0 + a) 2 (0 + /3) 2 = 0. 
The discriminant of this quartic function of 0 is a function of x, y, z, a, b, c of 
the degree 6 in (x, y, z) and 12 in (a, b, c); it contains however the factors xyz, a 2 b 2 c 2 , 
and the remaining factor is of the degree 3 in (x, y, z) and 6 in (a, b, c); this 
remaining factor is as will presently be seen 
= (a?x + b 2 y + c-zf - 27a 2 b 2 c 2 xyz. 
The last mentioned sextic equation in 0 will have a triple root 0 — — 8, if only 
the value 0 = — B makes to vanish the factor in [ J, that is if we have 
0 = g-h 2 x + h 2 f 2 y + fYz. 
of the sextic