ON A CERTAIN SEXTIC TORSE. 
Ill 
436] 
or observing that the coefficients of a 2 g 2 h 2 x 2 , b 2 h 2 fy and c 2 f 2 g 2 z 2 are equal to each other 
and to 
the equation becomes 
62p 2 q 2 r 2 — 28 ((f r r + r ? jr +p 3 q 3 ), 
(*) = — 3 (62p 2 q 2 r 2 - 28 (q 3 r 3 + r 3 p 3 + p 3 q 3 ) (a 2 g 2 h 2 x 2 + b 2 h 2 fy + c 2 / 2 g 2 z 2 ) 
+ 3 (3p s — 14p s qr + 130pyr 2 + 13 Qp 2 q 3 r 3 — 42 qh A )f 2 yz 
+ 3 (3q 8 — 14>q 6 pr + 130q*p 2 r 2 + 1 S6q 2 p 3 r 3 — 42?^p 4 ) g 2 zx 
+ 3 (3r 8 — 14r 6 pq + 130r*p 2 q 2 + 1S6r 2 p 3 q 3 — 42py) li 2 xy ; 
and we thence obtain by symmetry the complete value of (*\x, y, z, wf, viz. we have 
only to complete the literal parts of the foregoing expression into the forms 
o?g 2 h 2 x 2 + b 2 h 2 f 2 y 2 + c 2 f 2 g 2 z 2 + a?b 2 c 2 w 2 , 
f'-yz + a?xw, 
g 2 zx + bhyiu, 
c-xy + d 2 zw, 
respectively. 
20. The equation of the torse thus is 
A = a s g 6 h e x s + b 6 h 9 fy 4- c G f 6 g (i z 6 + a s b s chu s 
—f 6 (№ y + r f z y Q^y + & z y 
— g 6 (f 2 z + h 2 x) 3 (c 2 z + a 2 x ) s 
— h 6 (g 2 x + f 2 y) 3 + b 2 y ) 3 
— a 6 (g 2 x + c 2 w) 3 (h 2 x + bhu ) 3 
— 6 6 (h 2 y + ahuf (f 2 y + c 2 w) 3 
— c 6 ( f 2 z + bhu) 3 (g 2 z + a?w) 3 
+ (b 2 fhy + c 2 f 2 z + b 2 c 2 w ) 3 [(hhy +g 2 z + a?w) 3 - 27a?g 2 h? yzw ] 
+ {d 2 g 2 x + chfz + c 2 ahu) 3 [(f 2 z + h 2 x + bhvf - 27b 2 h 2 f 2 zxiu] 
+ (a 2 h 2 x + b 2 lihy + a 2 bhu) 3 [{g 2 x +f 2 y + chu) 3 - 27c 2 f 2 g 2 xyw] 
+ (g 2 h 2 x + h 2 f 2 y + f 2 g 2 zj [{a?x + bhj + c 2 * ) 3 - 27a 2 & 2 c 2 xyz ] 
+ xyzw (*][x, y, z, w) 2 = 0. 
I recall that 
a = /3-y, f = a-8, p = af=(cc-8)(/3 - y), 
b = y — a, g = /3 — 8, q = bg = (/3 — 8)(y — a), 
c = a — /3, h = y-8, r=ch = (y-8)(a-/3). 
Developing, we have finally the equation of the torse m the form following.