100 
ON A CERTAIN SEXTIC TORSE. 
[436 
anticipate the remark that the coordinates (x, y, z, w) of a point on the curve may 
by an obvious reduction be rendered proportional to the fourth powers (9 + a) 4 , (9 + (3f, 
(9 +<y) 4 , (Q + Sf in the parameter 9; this leads to an equation 
x , y 2 , w _ n 
W+*f + W+W + W+vf + ¥ + si 2 “ ’ 
for the osculating plane at the point (x, y, z, w) ; or observing that this equation, 
when integralised, is of the form (x, y, z, w\9, 1) 6 = 0, we see that the equation is 
obtained by equating to zero the discriminant of a certain sextic function in 9\ the 
discriminant is of the order 10 in the coordinates (x, y, z, w), but it obviously contains 
the factor xyzw, or throwing this out we have an equation of the order 6, so that 
the torse is (as above stated) a sextic torse. 
Theorem relating to Four Binary Quartics. 
1. Consider the four quartics: 
(«!, b 1} Ci, di, ef$x, yf, 
(a,, b. 2> Co, d 2 , e.&x, yf, 
(a 3 , b 3 , c 3 , d 3 , e 3 ffx, yf, 
(a 4 , 6 4 , c 4 , d^ e$x, yf, 
then if X.J, X 2 , \ 3 , X 4 are any four quantities, these may be determined, and that in 
four different ways, so that 
^(Oi, yf + X 2 (a 2 ,...Jx, yf + \ 3 (a 3 ,...\x, yf + X 4 (a 4 , yf = (J3x + ayf, 
a perfect fourth power ; in fact, equating the coefficients of the different powers 
of (x, yf, we have five equations, which determine the ratios of the unknown quantities 
X n X 2 , X 3 , X 4 ; a, /3: eliminating X u X 2 , X 3 , X 4 , we find the equation 
/3\ 
/3 3 a, 
/3 2 a 2 , 
/3a 3 , 
a 4 
= 0, 
Oi, 
b 1} 
^1 > 
di, 
a n, 
b 2 , 
C 2 , 
d 3 , 
c 2 
«3, 
b 3 , 
^3 > 
d 3 , 
a i} 
b 4 , 
C 4 , 
d 4 , 
e 4 
giving four different values of the ratio a : /5; or, assigning at pleasure a value to 
a or /3 (say /3 = 1), then to each of the four sets of values of (a, /3) there correspond 
a determinate set of values of (X 1? X 2 , X 3 , X 4 ); that is, we have as stated four sets of 
values of X } , X 2 , X 3 , X 4 ; a, /3.