436] 
ON A CERTAIN SEXTIC TORSE. 
105 
By symmetry, we conclude that the sections by the principal planes x — 0 : 
y = 0, z — 0, w — 0, are each made up of a line taken three times, and of a cubic 
curve: viz. these are 
x — 0, . b 2 f 2 y + c 2 f 2 z + b 2 c 2 w = 0, 
y = 0, a 2 g 2 x . c 2 g 2 z + c 2 a 2 w = 0, 
z — 0, a 2 h 2 x + Mi 2 y . + a%hu = 0, 
w = 0, g 2 K 2 x + ltf 2 y + f' 2 g 2 z . = 0, 
{b-y) A + Yfz)" + {ahvf = 0, 
(h-xf . + (fzf + {bhvf = 0, 
(g-xf + (fyf . + {chvf = 0, 
(a 2 xY + (b 2 yf + (c 2 zf - 0, 
where for shortness I have written the equations of the four cubics in their irrational 
forms respectively. 
Partial Determination of the Equation. 
10. As the value of A is known when any one of the coordinates x, y, z, w is 
put = 0, we in fact know all the terms of A, except those which contain the factor 
xyzio, which unknown terms, as A is of the degree 6, are of the form (* fx, y, z, wf. 
I remark that if (xyzw) is any homogeneous function (*$#, y, z, w) 2 , and (xyz), (xy), 
(x) are what {xyzw) become on putting therein {w = 0), (z = 0, w — 0), {y = 0, z = 0, w = 0) 
respectively, and the like for the other similar symbols, then that 
(xyzw) = {x) + {y) + {z) + {w) 
— {xy) — {xz) — {xw) — {yz) — (yw) — {zw) 
+ {xyz) + {xyw) + {xzw) + {yzw) 
+ terms multiplied by xyzw ; 
in fact, omitting the last line, this equation on writing therein x — 0 or y = 0 or z = O 
or w = 0, becomes an identity, that is, the difference of the two sides vanishes when 
any one of these equations is satisfied, and such difference contains therefore the factor 
xyzw ; which proves the theorem. It hence appears that the equation A = 0 of the 
torse is 
A = a ÿ g 6 h 6 x 6 + b 6 h s f 6 y 6 + c r f 6 g e z B + a 6 b B c B w B 
— {g 2 h 2 x + h 2 f' 2 yY {a-x + b‘ 2 yf 
— {k 2 f 2 y + fyz) 3 {b 2 y + c 2 z) 2 
— {g 2 li 2 x + f‘ 2 g' 2 z) s {a 2 x + c 2 z f 
— {a 2 Ii 2 x + a?b 2 wf {g 2 x + c 2 wf 
— {b' 2 h‘ 2 y + a?b 2 wf {f 2 y + c 2 w) 3 
— {c 2 g‘ 2 z + c 2 a 2 wf {f 2 z + bhv) 3 
+ {bf ' 2 y + c 2 f 2 z + b 2 c 2 wf [(% + g 2 z + a 2 w) s - 27a 2 g 2 h 2 yzw] 
+ {ayx + c 2 g 2 z + c 2 a 2 wy [{Ji 2 x +f 2 z + b 2 wf - 2lb 2 hf 2 zxvj] 
+ {a 2 k 2 x + b 2 h 2 y + urbhof {{g 2 x +f 2 y + c 2 wf - 21 cf y xyw] 
+ (ylrx + A 2 f 2 y + fyz) 3 [{a 2 x + b 2 y + c 2 z f - 2la 2 b 2 c 2 xyz ] 
+ xyzw) {*Yt x > V’ z > W Y’ 
where the ten coefficients of {*\x, y, z, w) 2 remain to be found. 
C. VII. 
14